Get answer: For the following exercises, solve the equation for x, if there is a

For the following exercises, use like bases to solve the exponential equation. \(4^{-3 v-2}=4^{-v}\) Text Transcription: 4^-3v-2 =4^-v

For the following exercises, use like bases to solve the exponential equation. \(64 \cdot 4^{3 x}=16\) Text Transcription: 64 \cdot 4^3x =16

For the following exercises, use like bases to solve the exponential equation. \(3^{2 x+1} \cdot 3^{x}=243\) Text Transcription: 3^{2x+1\cdot 3^x =243

For the following exercises, use like bases to solve the exponential equation. \(2^{-3 n} \cdot \frac{1}{4}=2^{n+2}\) Text Transcription: 2^-3 n \cdot \frac 1/4=2^n+2

For the following exercises, use like bases to solve the exponential equation. \(625 \cdot 5^{3 x+3}=125\) Text Transcription: 625 \cdot 5^3x+3 =125

For the following exercises, use like bases to solve the exponential equation. \(\frac{36^{3 b}}{36^{2 b}}=216^{2-b}\) Text Transcription: \frac 36^3b/36^2b =216^2-b

For the following exercises, use like bases to solve the exponential equation. \(\left(\frac{1}{64}\right)^{3 n} \cdot 8=2^{6}\) Text Transcription: (\frac 1/64)^3n \cdot 8=2^6

For the following exercises, use logarithms to solve. \(9^{x-10}=1\) Text Transcription: 9^x-10 =1

For the following exercises, use logarithms to solve. \(2 e^{6 x}=13\) Text Transcription: 2e^6x =13

For the following exercises, use logarithms to solve. \(e^{r+10}-10=-42\) Text Transcription: e^r+10 -10=-42

For the following exercises, use logarithms to solve. \(2 \cdot 10^{9 a}=29\) Text Transcription: 2\cdot 10^9a =29

For the following exercises, use logarithms to solve. \(-8 \cdot 10^{p+7}-7=-24\) Text Transcription: -8 \cdot 10^p+7 -7=-24

For the following exercises, use logarithms to solve. \(7 e^{3 n-5}+5=-89\) Text Transcription: 7e^3n-5 +5=-89

For the following exercises, use logarithms to solve. \(e^{-3 k}+6=44\) Text Transcription: e^-3k +6=44

For the following exercises, use logarithms to solve. \(-5 e^{9 x-8}-8=-62\) Text Transcription: -5e^9x-8 -8=-62

For the following exercises, use logarithms to solve. \(-6 e^{9 x+8}+2=-74\) Text Transcription: -6e^9x+8 +2=-74

For the following exercises, use logarithms to solve. \(2^{x+1}=5^{2 x-1}\) Text Transcription: 2^x+1 =5^2x-1

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-132=0\) Text Transcription: e^2x -e^x -132=0

For the following exercises, use logarithms to solve. \(7 e^{8 x+8}-5=-95\) Text Transcription: 7e^8x+8 -5=-95

For the following exercises, use logarithms to solve. \(10 e^{8 x+3}+2=8\) Text Transcription: 10e^8x+3 +2=8

For the following exercises, use logarithms to solve. \(4 e^{3 x+3}-7=53\) Text Transcription: 4e^3x+3 -7=53

For the following exercises, use logarithms to solve. \(8 e^{-5 x-2}-4=-90\) Text Transcription: 8e^-5x-2 -4=-90

For the following exercises, use logarithms to solve. \(3^{2 x+1}=7^{x-2}\) Text Transcription: 3^2x+1 =7^x-2

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-6=0\) Text Transcription: e^2x -e^x -6=0

For the following exercises, use logarithms to solve. \(3 e^{3-3 x}+6=-31\) Text Transcription: 3e^3-3x +6=-31

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log \left(\frac{1}{100}\right)=-2\) Text Transcription: log(\frac 1/100)=-2

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log _{324}(18)=\frac{1}{2}\) Text Transcription: log_324 (18)=\frac 1/2

For the following exercises, use the definition of a logarithm to solve the equation. \(5 \log _{7}(n)=10\) Text Transcription: 5log_7 (n)=10

For the following exercises, use the definition of a logarithm to solve the equation. \(-8 \log _{9}(x)=16\) Text Transcription: -8log_9 (x)=16

For the following exercises, use the definition of a logarithm to solve the equation. \(4+\log _{2}(9 k)=2\) Text Transcription: 4+log_2 (9k)=2

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{13}(5 n-2)=\log _{13}(8-5 n)\) Text Transcription: log_13 (5n-2)=\log_13 (8-5n)

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln (-3 x)=\ln \left(x^{2}-6 x\right)\) Text Transcription: ln(-3x)=ln(x^2 -6x)

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{4}(6-m)=\log _{4} 3(m)\) Text Transcription: log_4 (6-m)=log_4 3(m)

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{9}\left(2 n^{2}-14 n\right)=\log _{9}\left(-45+n^{2}\right)\) Text Transcription: log_9 (2n^2 -14n)=log_9 (-45+n^2)

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln \left(x^{2}-10\right)+\ln (9)=\ln (10)\) Text Transcription: ln(x^2 -10)+ln(9)=ln(10)

For the following exercises, solve each equation for x. \(\log _{2}(7 x+6)=3\) Text Transcription: log_2 (7x+6)=3

For the following exercises, solve each equation for x. \(\ln (7)+\ln \left(2-4 x^{2}\right)=\ln (14)\) Text Transcription: ln(7)+ln(2-4x^2)=ln(14)

For the following exercises, solve each equation for x. \(\log _{8}(x+6)-\log _{8}(x)=\log _{8}(58)\) Text Transcription: log_8 (x+6)-log_8 (x)=log_8 (58)

For the following exercises, solve each equation for x. \(\log _{3}(3 x)-\log _{3}(6)=\log _{3}(77)\) Text Transcription: log_3 (3x)-log_3 (6)=log_3 (77)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(x)-5=-4\) Text Transcription: log_9 (x)-5=-4

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{3}(x)+3=2\) Text Transcription: log_3 (x)+3=2

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(-7+\log _{3}(4-x)=-6\) Text Transcription: -7+log_3 (4-x)=-6

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{11}\left(-2 x^{2}-7 x\right)=\log _{11}(x-2)\) Text Transcription: log_11 (-2x^2 -7x)=log_11 (x-2)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(3-x)=\log _{9}(4 x-8)\) Text Transcription: log_9 (3-x)=log_9 (4 x-8)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log \left(x^{2}+13\right)=\log (7 x+3)\) Text Transcription: log(x^2 +13)=log(7x+3)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\frac{3}{\log _{2}(10)}-\log (x-9)=\log (44)\) Text transcription: \frac 3/log_2 (10) -log(x-9)=log(44)

The formula for measuring sound intensity in decibels D is defined by the equation \(D=10 \log \left(\frac{I}{I_{0}}\right)\), where I is the intensity of the sound in watts per square meter and \(I_{0}=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of \(8.3 \cdot 10^{2}\) watts per square meter? Text Transcription: D=10 log(\frac I/I_0) I_0 =10^-12 8.3 \cdot10^2

The population of a small town is modeled by the equation \(P=1650 e^{0.5 t}\) where t is measured in years. In approximately how many years will the town’s population reach 20,000? Text Transcription: P=1650e^0.5t

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(1000(1.03)^{t}=5000\) using the common log. Text Transcription: 1000(1.03)^t=5000

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(e^{5 x}=17\) using the natural log Text Transcription: e^5x =17

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3(1.04)^{3 t}=8\) using the common log Text Transcription: 3(1.04)^3t =8

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3^{4 x-5}=38\) using the common log Text Transcription: 3^4x-5 =38

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(50 e^{-0.12 t}=10\) using the natural log Text Transcription: 50e^-0.12t =10

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. \(7 e^{3 x-5}+7.9=47\) Text Transcription: 7e^3x-5 +7.9=47

Atmospheric pressure P in pounds per square inch is represented by the formula \(P=14.7 e^{-0.21 x}\), where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile) Text Transcription: P=14.7e^-0.21x

The magnitude M of an earthquake is represented by the equation \(M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\) where E is the amount of energy released by the earthquake in joules and \(E_{0}=10^{4.4}\) is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing \(1.4 \cdot 10^{13}\) joules of energy? Text Transcription: M=\frac 2/3 log(\frac E/E_0) E_0 =10^4.4 1.4 \cdot10^13

Use the definition of a logarithm along with the oneto-one property of logarithms to prove that \(b^{\log _{b} x}=x\). Text Transcription: b^log_b x =x

Recall the formula for continually compounding interest, \(y=A e^{k t}\). Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. Text Transcription: y=Ae^kt

Recall the compound interest formula \(A=a\left(1+\frac{r}{k}\right)^{k t}\). Use the definition of a logarithm along with properties of logarithms to solve the formula for time t. Text Transcription: A=a(1+\frac r/k)^kt

Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the equation \(T=T_{s}+\left(T_{0}-T_{s}\right) e^{-k t}\), where Ts is the temperature of the surrounding environment, T0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. Text Transcription: T=T_s +(T_0 -T_s)e^-kt

For the following exercises, use like bases to solve the exponential equation. \(4^{-3 v-2}=4^{-v}\) Text Transcription: 4^-3v-2 =4^-v

For the following exercises, use like bases to solve the exponential equation. \(64 \cdot 4^{3 x}=16\) Text Transcription: 64 \cdot 4^3x =16

For the following exercises, use like bases to solve the exponential equation. \(3^{2 x+1} \cdot 3^{x}=243\) Text Transcription: 3^{2x+1\cdot 3^x =243

For the following exercises, use like bases to solve the exponential equation. \(2^{-3 n} \cdot \frac{1}{4}=2^{n+2}\) Text Transcription: 2^-3 n \cdot \frac 1/4=2^n+2

For the following exercises, use like bases to solve the exponential equation. \(625 \cdot 5^{3 x+3}=125\) Text Transcription: 625 \cdot 5^3x+3 =125

For the following exercises, use like bases to solve the exponential equation. \(\frac{36^{3 b}}{36^{2 b}}=216^{2-b}\) Text Transcription: \frac 36^3b/36^2b =216^2-b

For the following exercises, use like bases to solve the exponential equation. \(\left(\frac{1}{64}\right)^{3 n} \cdot 8=2^{6}\) Text Transcription: (\frac 1/64)^3n \cdot 8=2^6

For the following exercises, use logarithms to solve. \(9^{x-10}=1\) Text Transcription: 9^x-10 =1

For the following exercises, use logarithms to solve. \(2 e^{6 x}=13\) Text Transcription: 2e^6x =13

For the following exercises, use logarithms to solve. \(e^{r+10}-10=-42\) Text Transcription: e^r+10 -10=-42

For the following exercises, use logarithms to solve. \(2 \cdot 10^{9 a}=29\) Text Transcription: 2\cdot 10^9a =29

For the following exercises, use logarithms to solve. \(-8 \cdot 10^{p+7}-7=-24\) Text Transcription: -8 \cdot 10^p+7 -7=-24

For the following exercises, use logarithms to solve. \(7 e^{3 n-5}+5=-89\) Text Transcription: 7e^3n-5 +5=-89

For the following exercises, use logarithms to solve. \(e^{-3 k}+6=44\) Text Transcription: e^-3k +6=44

For the following exercises, use logarithms to solve. \(-5 e^{9 x-8}-8=-62\) Text Transcription: -5e^9x-8 -8=-62

For the following exercises, use logarithms to solve. \(-6 e^{9 x+8}+2=-74\) Text Transcription: -6e^9x+8 +2=-74

For the following exercises, use logarithms to solve. \(2^{x+1}=5^{2 x-1}\) Text Transcription: 2^x+1 =5^2x-1

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-132=0\) Text Transcription: e^2x -e^x -132=0

For the following exercises, use logarithms to solve. \(7 e^{8 x+8}-5=-95\) Text Transcription: 7e^8x+8 -5=-95

For the following exercises, use logarithms to solve. \(10 e^{8 x+3}+2=8\) Text Transcription: 10e^8x+3 +2=8

For the following exercises, use logarithms to solve. \(4 e^{3 x+3}-7=53\) Text Transcription: 4e^3x+3 -7=53

For the following exercises, use logarithms to solve. \(8 e^{-5 x-2}-4=-90\) Text Transcription: 8e^-5x-2 -4=-90

For the following exercises, use logarithms to solve. \(3^{2 x+1}=7^{x-2}\) Text Transcription: 3^2x+1 =7^x-2

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-6=0\) Text Transcription: e^2x -e^x -6=0

For the following exercises, use logarithms to solve. \(3 e^{3-3 x}+6=-31\) Text Transcription: 3e^3-3x +6=-31

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log \left(\frac{1}{100}\right)=-2\) Text Transcription: log(\frac 1/100)=-2

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log _{324}(18)=\frac{1}{2}\) Text Transcription: log_324 (18)=\frac 1/2

For the following exercises, use the definition of a logarithm to solve the equation. \(5 \log _{7}(n)=10\) Text Transcription: 5log_7 (n)=10

For the following exercises, use the definition of a logarithm to solve the equation. \(-8 \log _{9}(x)=16\) Text Transcription: -8log_9 (x)=16

For the following exercises, use the definition of a logarithm to solve the equation. \(4+\log _{2}(9 k)=2\) Text Transcription: 4+log_2 (9k)=2

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{13}(5 n-2)=\log _{13}(8-5 n)\) Text Transcription: log_13 (5n-2)=\log_13 (8-5n)

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln (-3 x)=\ln \left(x^{2}-6 x\right)\) Text Transcription: ln(-3x)=ln(x^2 -6x)

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{4}(6-m)=\log _{4} 3(m)\) Text Transcription: log_4 (6-m)=log_4 3(m)

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{9}\left(2 n^{2}-14 n\right)=\log _{9}\left(-45+n^{2}\right)\) Text Transcription: log_9 (2n^2 -14n)=log_9 (-45+n^2)

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln \left(x^{2}-10\right)+\ln (9)=\ln (10)\) Text Transcription: ln(x^2 -10)+ln(9)=ln(10)

For the following exercises, solve each equation for x. \(\log _{2}(7 x+6)=3\) Text Transcription: log_2 (7x+6)=3

For the following exercises, solve each equation for x. \(\ln (7)+\ln \left(2-4 x^{2}\right)=\ln (14)\) Text Transcription: ln(7)+ln(2-4x^2)=ln(14)

For the following exercises, solve each equation for x. \(\log _{8}(x+6)-\log _{8}(x)=\log _{8}(58)\) Text Transcription: log_8 (x+6)-log_8 (x)=log_8 (58)

For the following exercises, solve each equation for x. \(\log _{3}(3 x)-\log _{3}(6)=\log _{3}(77)\) Text Transcription: log_3 (3x)-log_3 (6)=log_3 (77)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(x)-5=-4\) Text Transcription: log_9 (x)-5=-4

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{3}(x)+3=2\) Text Transcription: log_3 (x)+3=2

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(-7+\log _{3}(4-x)=-6\) Text Transcription: -7+log_3 (4-x)=-6

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{11}\left(-2 x^{2}-7 x\right)=\log _{11}(x-2)\) Text Transcription: log_11 (-2x^2 -7x)=log_11 (x-2)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(3-x)=\log _{9}(4 x-8)\) Text Transcription: log_9 (3-x)=log_9 (4 x-8)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log \left(x^{2}+13\right)=\log (7 x+3)\) Text Transcription: log(x^2 +13)=log(7x+3)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\frac{3}{\log _{2}(10)}-\log (x-9)=\log (44)\) Text transcription: \frac 3/log_2 (10) -log(x-9)=log(44)

The formula for measuring sound intensity in decibels D is defined by the equation \(D=10 \log \left(\frac{I}{I_{0}}\right)\), where I is the intensity of the sound in watts per square meter and \(I_{0}=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of \(8.3 \cdot 10^{2}\) watts per square meter? Text Transcription: D=10 log(\frac I/I_0) I_0 =10^-12 8.3 \cdot10^2

The population of a small town is modeled by the equation \(P=1650 e^{0.5 t}\) where t is measured in years. In approximately how many years will the town’s population reach 20,000? Text Transcription: P=1650e^0.5t

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(1000(1.03)^{t}=5000\) using the common log. Text Transcription: 1000(1.03)^t=5000

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(e^{5 x}=17\) using the natural log Text Transcription: e^5x =17

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3(1.04)^{3 t}=8\) using the common log Text Transcription: 3(1.04)^3t =8

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3^{4 x-5}=38\) using the common log Text Transcription: 3^4x-5 =38

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(50 e^{-0.12 t}=10\) using the natural log Text Transcription: 50e^-0.12t =10

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. \(7 e^{3 x-5}+7.9=47\) Text Transcription: 7e^3x-5 +7.9=47

Atmospheric pressure P in pounds per square inch is represented by the formula \(P=14.7 e^{-0.21 x}\), where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile) Text Transcription: P=14.7e^-0.21x

The magnitude M of an earthquake is represented by the equation \(M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\) where E is the amount of energy released by the earthquake in joules and \(E_{0}=10^{4.4}\) is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing \(1.4 \cdot 10^{13}\) joules of energy? Text Transcription: M=\frac 2/3 log(\frac E/E_0) E_0 =10^4.4 1.4 \cdot10^13

Use the definition of a logarithm along with the oneto-one property of logarithms to prove that \(b^{\log _{b} x}=x\). Text Transcription: b^log_b x =x

Recall the formula for continually compounding interest, \(y=A e^{k t}\). Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. Text Transcription: y=Ae^kt

Recall the compound interest formula \(A=a\left(1+\frac{r}{k}\right)^{k t}\). Use the definition of a logarithm along with properties of logarithms to solve the formula for time t. Text Transcription: A=a(1+\frac r/k)^kt

Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the equation \(T=T_{s}+\left(T_{0}-T_{s}\right) e^{-k t}\), where Ts is the temperature of the surrounding environment, T0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. Text Transcription: T=T_s +(T_0 -T_s)e^-kt

For the following exercises, use like bases to solve the exponential equation. \(4^{-3 v-2}=4^{-v}\) Text Transcription: 4^-3v-2 =4^-v

For the following exercises, use like bases to solve the exponential equation. \(64 \cdot 4^{3 x}=16\) Text Transcription: 64 \cdot 4^3x =16

For the following exercises, use like bases to solve the exponential equation. \(3^{2 x+1} \cdot 3^{x}=243\) Text Transcription: 3^{2x+1\cdot 3^x =243

For the following exercises, use like bases to solve the exponential equation. \(2^{-3 n} \cdot \frac{1}{4}=2^{n+2}\) Text Transcription: 2^-3 n \cdot \frac 1/4=2^n+2

For the following exercises, use like bases to solve the exponential equation. \(625 \cdot 5^{3 x+3}=125\) Text Transcription: 625 \cdot 5^3x+3 =125

For the following exercises, use like bases to solve the exponential equation. \(\frac{36^{3 b}}{36^{2 b}}=216^{2-b}\) Text Transcription: \frac 36^3b/36^2b =216^2-b

For the following exercises, use like bases to solve the exponential equation. \(\left(\frac{1}{64}\right)^{3 n} \cdot 8=2^{6}\) Text Transcription: (\frac 1/64)^3n \cdot 8=2^6

For the following exercises, use logarithms to solve. \(9^{x-10}=1\) Text Transcription: 9^x-10 =1

For the following exercises, use logarithms to solve. \(2 e^{6 x}=13\) Text Transcription: 2e^6x =13

For the following exercises, use logarithms to solve. \(e^{r+10}-10=-42\) Text Transcription: e^r+10 -10=-42

For the following exercises, use logarithms to solve. \(2 \cdot 10^{9 a}=29\) Text Transcription: 2\cdot 10^9a =29

For the following exercises, use logarithms to solve. \(-8 \cdot 10^{p+7}-7=-24\) Text Transcription: -8 \cdot 10^p+7 -7=-24

For the following exercises, use logarithms to solve. \(7 e^{3 n-5}+5=-89\) Text Transcription: 7e^3n-5 +5=-89

For the following exercises, use logarithms to solve. \(e^{-3 k}+6=44\) Text Transcription: e^-3k +6=44

For the following exercises, use logarithms to solve. \(-5 e^{9 x-8}-8=-62\) Text Transcription: -5e^9x-8 -8=-62

For the following exercises, use logarithms to solve. \(-6 e^{9 x+8}+2=-74\) Text Transcription: -6e^9x+8 +2=-74

For the following exercises, use logarithms to solve. \(2^{x+1}=5^{2 x-1}\) Text Transcription: 2^x+1 =5^2x-1

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-132=0\) Text Transcription: e^2x -e^x -132=0

For the following exercises, use logarithms to solve. \(7 e^{8 x+8}-5=-95\) Text Transcription: 7e^8x+8 -5=-95

For the following exercises, use logarithms to solve. \(10 e^{8 x+3}+2=8\) Text Transcription: 10e^8x+3 +2=8

For the following exercises, use logarithms to solve. \(4 e^{3 x+3}-7=53\) Text Transcription: 4e^3x+3 -7=53

For the following exercises, use logarithms to solve. \(8 e^{-5 x-2}-4=-90\) Text Transcription: 8e^-5x-2 -4=-90

For the following exercises, use logarithms to solve. \(3^{2 x+1}=7^{x-2}\) Text Transcription: 3^2x+1 =7^x-2

For the following exercises, use logarithms to solve. \(e^{2 x}-e^{x}-6=0\) Text Transcription: e^2x -e^x -6=0

For the following exercises, use logarithms to solve. \(3 e^{3-3 x}+6=-31\) Text Transcription: 3e^3-3x +6=-31

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log \left(\frac{1}{100}\right)=-2\) Text Transcription: log(\frac 1/100)=-2

For the following exercises, use the definition of a logarithm to rewrite the equation as an exponential equation. \(\log _{324}(18)=\frac{1}{2}\) Text Transcription: log_324 (18)=\frac 1/2

For the following exercises, use the definition of a logarithm to solve the equation. \(5 \log _{7}(n)=10\) Text Transcription: 5log_7 (n)=10

For the following exercises, use the definition of a logarithm to solve the equation. \(-8 \log _{9}(x)=16\) Text Transcription: -8log_9 (x)=16

For the following exercises, use the definition of a logarithm to solve the equation. \(4+\log _{2}(9 k)=2\) Text Transcription: 4+log_2 (9k)=2

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{13}(5 n-2)=\log _{13}(8-5 n)\) Text Transcription: log_13 (5n-2)=\log_13 (8-5n)

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln (-3 x)=\ln \left(x^{2}-6 x\right)\) Text Transcription: ln(-3x)=ln(x^2 -6x)

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{4}(6-m)=\log _{4} 3(m)\) Text Transcription: log_4 (6-m)=log_4 3(m)

For the following exercises, use the one-to-one property of logarithms to solve. \(\log _{9}\left(2 n^{2}-14 n\right)=\log _{9}\left(-45+n^{2}\right)\) Text Transcription: log_9 (2n^2 -14n)=log_9 (-45+n^2)

For the following exercises, use the one-to-one property of logarithms to solve. \(\ln \left(x^{2}-10\right)+\ln (9)=\ln (10)\) Text Transcription: ln(x^2 -10)+ln(9)=ln(10)

For the following exercises, solve each equation for x. \(\log _{2}(7 x+6)=3\) Text Transcription: log_2 (7x+6)=3

For the following exercises, solve each equation for x. \(\ln (7)+\ln \left(2-4 x^{2}\right)=\ln (14)\) Text Transcription: ln(7)+ln(2-4x^2)=ln(14)

For the following exercises, solve each equation for x. \(\log _{8}(x+6)-\log _{8}(x)=\log _{8}(58)\) Text Transcription: log_8 (x+6)-log_8 (x)=log_8 (58)

For the following exercises, solve each equation for x. \(\log _{3}(3 x)-\log _{3}(6)=\log _{3}(77)\) Text Transcription: log_3 (3x)-log_3 (6)=log_3 (77)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(x)-5=-4\) Text Transcription: log_9 (x)-5=-4

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{3}(x)+3=2\) Text Transcription: log_3 (x)+3=2

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(-7+\log _{3}(4-x)=-6\) Text Transcription: -7+log_3 (4-x)=-6

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{11}\left(-2 x^{2}-7 x\right)=\log _{11}(x-2)\) Text Transcription: log_11 (-2x^2 -7x)=log_11 (x-2)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log _{9}(3-x)=\log _{9}(4 x-8)\) Text Transcription: log_9 (3-x)=log_9 (4 x-8)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\log \left(x^{2}+13\right)=\log (7 x+3)\) Text Transcription: log(x^2 +13)=log(7x+3)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. \(\frac{3}{\log _{2}(10)}-\log (x-9)=\log (44)\) Text transcription: \frac 3/log_2 (10) -log(x-9)=log(44)

The formula for measuring sound intensity in decibels D is defined by the equation \(D=10 \log \left(\frac{I}{I_{0}}\right)\), where I is the intensity of the sound in watts per square meter and \(I_{0}=10^{-12}\) is the lowest level of sound that the average person can hear. How many decibels are emitted from a jet plane with a sound intensity of \(8.3 \cdot 10^{2}\) watts per square meter? Text Transcription: D=10 log(\frac I/I_0) I_0 =10^-12 8.3 \cdot10^2

The population of a small town is modeled by the equation \(P=1650 e^{0.5 t}\) where t is measured in years. In approximately how many years will the town’s population reach 20,000? Text Transcription: P=1650e^0.5t

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(1000(1.03)^{t}=5000\) using the common log. Text Transcription: 1000(1.03)^t=5000

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(e^{5 x}=17\) using the natural log Text Transcription: e^5x =17

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3(1.04)^{3 t}=8\) using the common log Text Transcription: 3(1.04)^3t =8

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(3^{4 x-5}=38\) using the common log Text Transcription: 3^4x-5 =38

For the following exercises, solve each equation by rewriting the exponential expression using the indicated logarithm. Then use a calculator to approximate the variable to 3 decimal places. \(50 e^{-0.12 t}=10\) using the natural log Text Transcription: 50e^-0.12t =10

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. \(7 e^{3 x-5}+7.9=47\) Text Transcription: 7e^3x-5 +7.9=47

Atmospheric pressure P in pounds per square inch is represented by the formula \(P=14.7 e^{-0.21 x}\), where x is the number of miles above sea level. To the nearest foot, how high is the peak of a mountain with an atmospheric pressure of 8.369 pounds per square inch? (Hint: there are 5280 feet in a mile) Text Transcription: P=14.7e^-0.21x

The magnitude M of an earthquake is represented by the equation \(M=\frac{2}{3} \log \left(\frac{E}{E_{0}}\right)\) where E is the amount of energy released by the earthquake in joules and \(E_{0}=10^{4.4}\) is the assigned minimal measure released by an earthquake. To the nearest hundredth, what would the magnitude be of an earthquake releasing \(1.4 \cdot 10^{13}\) joules of energy? Text Transcription: M=\frac 2/3 log(\frac E/E_0) E_0 =10^4.4 1.4 \cdot10^13

Use the definition of a logarithm along with the oneto-one property of logarithms to prove that \(b^{\log _{b} x}=x\). Text Transcription: b^log_b x =x

Recall the formula for continually compounding interest, \(y=A e^{k t}\). Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. Text Transcription: y=Ae^kt

Recall the compound interest formula \(A=a\left(1+\frac{r}{k}\right)^{k t}\). Use the definition of a logarithm along with properties of logarithms to solve the formula for time t. Text Transcription: A=a(1+\frac r/k)^kt

Newton’s Law of Cooling states that the temperature T of an object at any time t can be described by the equation \(T=T_{s}+\left(T_{0}-T_{s}\right) e^{-k t}\), where Ts is the temperature of the surrounding environment, T0 is the initial temperature of the object, and k is the cooling rate. Use the definition of a logarithm along with properties of logarithms to solve the formula for time t such that t is equal to a single logarithm. Text Transcription: T=T_s +(T_0 -T_s)e^-kt

How can an exponential equation be solved?

When does an extraneous solution occur? How can an extraneous solution be recognized?

When can the one-to-one property of logarithms be used to solve an equation? When can it not be used?

For the following exercises, use the definition of a logarithm to solve the equation. 2log(8n + 4) + 6 = 10

For the following exercises, use the definition of a logarithm to solve the equation. 10 ? 4ln(9 ? 8x) = 6

For the following exercises, use the one-to-one property of logarithms to solve. ln(10 ? 3x) = ln(?4x)

For the following exercises, use the one-to-one property of logarithms to solve. log(x + 3) ? log(x) = log(74)

For the following exercises, use the one-to-one property of logarithms to solve. ln(x ? 2) ? ln(x) = ln(54)

For the following exercises, solve each equation for x. log(x + 12) = log(x) + log(12)

For the following exercises, solve each equation for x. ln(x) + ln(x ? 3) = ln(7x)

For the following exercises, solve each equation for x. ln(3) ? ln(3 ? 3x) = ln(4)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. ln(3x) = 2

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. ln(x ? 5) = 1

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. log(4) + log(?5x) = 2

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. ln(4x ? 10) ? 6 = ?5

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. log(4 ? 2x) = log(?4x)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. ln(2x + 9) = ln(?5x)

For the following exercises, solve the equation for x, if there is a solution. Then graph both sides of the equation, and observe the point of intersection (if it exists) to verify the solution. ln(x) ? ln(x + 3) = ln(6)

An account with an initial deposit of $6,500 earns 7.25% annual interest, compounded continuously. How much will the account be worth after 20 years?

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. ln(3) + ln(4.4x + 6.8) = 2

For the following exercises, use a calculator to solve the equation. Unless indicated otherwise, round all answers to the nearest ten-thousandth. log(?0.7x ? 9) = 1 + 5log(5)