In Exercises 131 and 132, find the exact value of and then approximate using Stirlings

Evaluating an Expression In Exercises 1 and 2, evaluate the expressions. (a) \(25^{3 / 2}\) (b) \(81^{1 / 2}\) (c) \(3^{-2}\) (d) \(27^{-1 / 3}\) Text Transcription: 25^{3 / 2} 81^{1 / 2} 3^{-2} 27^{-1 / 3}

Evaluating an Expression In Exercises 1 and 2, evaluate the expressions. (a) \(64^{1 / 3}\) (b) \(5^{-4}\) (c) \(\left(\frac{1}{8}\right)^{1 / 3}\) (d) \(\left(\frac{1}{4}\right)^{3}\) Text Transcription: 64^{1 / 3} 5^{-4} (1 / 8)^{1 / 3} (1 / 4)^{3}

In Exercises 3-6, use the properties of exponents to simplify the expressions. (a) \(\left(5^{2}\right)\left(5^{3}\right)\) (b) \(\left(5^{2}\right)\left(5^{-3}\right)\) (c) \(\frac{5^{3}}{25^{2}}\) (d) \(\left(\frac{1}{4}\right)^{2} 2^{6}\) Text Transcription: (5^{2})(5^{3}) (5^{2})(5^{-3}) 5^{3} / {25^{2} (1 / 4)^{2} 2^{6}

In Exercises 3-6, use the properties of exponents to simplify the expressions. (a) \(\left(2^{2}\right)^{3}\) (b) \(\left(5^{4}\right)^{1 / 2}\) (c) \(\left[\left(27^{-1}\right)\left(27^{2 / 3}\right)\right]^{3}\) (d) \(\left(25^{3 / 2}\right)\left(3^{2}\right)\) Text Transcription: (2^{2})^{3} (5^{4}^{1 / 2} [(27^{-1})(27^{2 / 3})]^{3} (25^{3 / 2})(3^{2})

In Exercises 3-6, use the properties of exponents to simplify the expressions. (a) \(e^{2}\left(e^{4}\right)\) (b) \(\left(e^{3}\right)^{4}\) (c) \(\left(e^{3}\right)^{-2}\) (d) \(\frac{e^{5}}{e^{3}}\) Text Transcription: e^{2}(e^{4}) (e^{3})^{4} (e^{3})^{-2} e^{5} / e^{3}

In Exercises 3-6, use the properties of exponents to simplify the expressions. (a) \(\left(\frac{1}{e}\right)^{-2}\) (b) \(\left(\frac{e^{5}}{e^{2}}\right)^{-1}\) (c) \(e^{0}\) (d) \(\frac{1}{e^{-3}}\) Text Transcription: (1 / e)^{-2} ({e^{5} / {e^{2})^{-1} e^{0} 1 / e^{-3}

In Exercises 7-22, solve for \(x\). \(3^{x}=81\) Text Transcription: x 3^x = 81

In Exercises 7-22, solve for \(x\). \(4^{x}=64\) Text Transcription: x 4^{x} = 64

In Exercises 7-22, solve for \(x\). \(6^{x-2}=36\) Text Transcription: x 6^{x - 2} = 36

In Exercises 7-22, solve for \(x\). \(5^{x+1}=125\) Text Transcription: x 5^{x + 1} = 125

In Exercises 7-22, solve for \(x\). \(\left(\frac{1}{2}\right)^{x}=32\) Text Transcription: x (1 / 2)^{x} = 32

In Exercises 7-22, solve for \(x\). \(\left(\frac{1}{4}\right)^{x}=16\) Text Transcription: x (1 / 4)^{x} = 16

In Exercises 7-22, solve for \(x\). \(\left(\frac{1}{3}\right)^{x-1}=27\) Text Transcription: x (1 / 3)^{x - 1} = 27

In Exercises 7-22, solve for \(x\). \(\left(\frac{1}{5}\right)^{2 x}=625\) Text Transcription: x (1 / 5)^{2x} = 625

In Exercises 7-22, solve for \(x\). \(4^{3}=(x+2)^{3}\) Text Transcription: x 4^{3} = (x + 2)^3

In Exercises 7-22, solve for \(x\). \(18^{2}=(5 x-7)^{2}\) Text Transcription: x 18^{2} = (5 x - 7)^2

In Exercises 7-22, solve for \(x\). \(x^{3 / 4}=8\) Text Transcription: x x^{3 / 4} = 8

In Exercises 7-22, solve for \(x\). \((x+3)^{4 / 3}=16\) Text Transcription: x (x + 3)^{4 / 3} = 16

In Exercises 7-22, solve for \(x\). \(e^{x}=5\) Text Transcription: x e^{x} = 5

In Exercises 7-22, solve for \(x\). \(e^{x}=1\) Text Transcription: x e^{x} = 1

In Exercises 7-22, solve for \(x\). \(e^{-2 x}=e^{5}\) Text Transcription: x e^{ -2x} = e^{5}

In Exercises 7-22, solve for \(x\). \(e^{3 x}=e^{-4}\) Text Transcription: x e^{3x} = e^{-4}

In Exercises 23 and 24, compare the given number with the number \(e\). Is the number less than or greater than \(e\)? \((1+\frac{1}{1,000,000}\right)^{1,000,000}\) Text Transcription: e (1 {1} / {1,000,000})^1,000,000

In Exercises 23 and 24, compare the given number with the number \(e\). Is the number less than or greater than \(e\)? \(1+1+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+\frac{1}{720}+\frac{1}{5040}\) Text Transcription: e 1 + 1 + 1 / 2 + 1 / 6 + 1 / 24 + 1 / 120 + 1 / 720 + 1 / 5040

In Exercises 25-38, sketch the graph of the function. \(y=3^{x}\) Text Transcription: y = 3^{x}

In Exercises 25-38, sketch the graph of the function. \(y=3^{x-1}\) Text Transcription: y = 3^{x - 1}

In Exercises 25-38, sketch the graph of the function. \(y=\left(\frac{1}{3}\right)^{x}\) Text Transcription: y = (1 / 3)^{x}

In Exercises 25-38, sketch the graph of the function. \(y=2^{-x^{2}}\) Text Transcription: y = 2^{-x^2}

In Exercises 25-38, sketch the graph of the function. \(f(x)=3^{-x^{2}}\) Text Transcription: f(x) = 3^{-x^2}

In Exercises 25-38, sketch the graph of the function. \(f(x)=3^{|x|}\) Text Transcription: f(x) = 3^|x|

In Exercises 25-38, sketch the graph of the function. \(y=e^{-x}\) Text Transcription: y = e^-x

In Exercises 25-38, sketch the graph of the function. \(y=\frac{1}{2} e^{x}\) Text Transcription: \(y = 1 / 2 e^{x}\)

In Exercises 25-38, sketch the graph of the function. \(y=e^{x}+2\) Text Transcription: y = e^x + 2

In Exercises 25-38, sketch the graph of the function. \(y=e^{x-1}\) Text Transcription: y = e^x - 1

In Exercises 25-38, sketch the graph of the function. \(h(x)=e^{x-2}\) Text Transcription: h(x) = e^x - 2

In Exercises 25-38, sketch the graph of the function. \(g(x)=-e^{x / 2}\) Text Transcription: g(x) = -e^x / 2

In Exercises 25-38, sketch the graph of the function. \(y=e^{-x^{2}}\) Text Transcription: y = e^{-x^2}

In Exercises 25-38, sketch the graph of the function. \(y=e^{-x / 4}\) Text Transcription: y = e^{-x / 4}

In Exercises 39-44, find the domain of the function. \(f(x)=\frac{1}{3+e^{x}}\) Text Transcription: f(x) = 1 / 3 + e^x

In Exercises 39-44, find the domain of the function. \(f(x)=\frac{1}{2-e^{x}}\) Text Transcription: f(x) = 1 / 2 - e^x

In Exercises 39-44, find the domain of the function. \(f(x)=\sqrt{1-4^{x}}\) Text Transcription: f(x) = sqrt{1 - 4^x

In Exercises 39-44, find the domain of the function. \(f(x)=\sqrt{1+3^{-x}}\) Text Transcription: f(x) = sqrt{1 + 3^-x}

In Exercises 39-44, find the domain of the function. \(f(x)=\sin e^{-x}\) Text Transcription: f(x) = sin e^-x

In Exercises 39-44, find the domain of the function. \(f(x)=\cos e^{-x}\) Text Transcription: f(x) = cos e^-x

Use a graphing utility to graph \(f(x)=e^{x}\) and the given function in the same viewing window. How are the two graphs related? (a) \(g(x)=e^{x-2}\) (b) \(h(x)=-\frac{1}{2} e^{x}\) (c) \(q(x)=e^{-x}+3\) Text Transcription: f(x) = e^x g(x) = e^x - 2 h(x) = -1 / 2 e^x q(x) = e^-x + 3

Use a graphing utility to graph the function. Describe the shape of the graph for very large and very small values of \(x\). (a) \(f(x)=\frac{8}{1+e^{-0.5 x}}\) (b) \(g(x)=\frac{8}{1+e^{-0.5 / x}}\) Text Transcription: x f(x) = 8 / 1 + e^-0.5x g(x) = 8 / 1 + e^-0.5 / x

In Exercises 47-50, match the equation with the correct graph. Assume that \(a\) and \(C\) are positive real numbers. [The graphs are labeled (a), (b), (c), and (d).] . \(y=C e^{a x}\) Text Transcription: .y = Ce^{ax}

In Exercises 47-50, match the equation with the correct graph. Assume that \(a\) and \(C\) are positive real numbers. [The graphs are labeled (a), (b), (c), and (d).] \(y=C e^{-a x}\) Text Transcription: y = Ce^-ax

In Exercises 47-50, match the equation with the correct graph. Assume that \(a\) and \(C\) are positive real numbers. [The graphs are labeled (a), (b), (c), and (d).] \(y=C\left(1-e^{-a x}\right)\) Text Transcription: y = C(1- e^{-ax})

In Exercises 47-50, match the equation with the correct graph. Assume that \(a\) and \(C\) are positive real numbers. [The graphs are labeled (a), (b), (c), and (d).] \(y=\frac{C}{1+e^{-a x}}\) Text Transcription: y = C / 1 + e^{-ax}

In Exercises 51 and 52, find the exponential function \(y=C a^{x}\) that fits the graph. Text Transcription: y = Ca^x

In Exercises 51 and 52, find the exponential function \(y=C a^{x}\) that fits the graph. Text Transcription: y = Ca^x

In Exercises 53–56, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x) = ln x + 1

In Exercises 53–56, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x) = - ln x

In Exercises 53–56, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x) = ln (x - 1)

In Exercises 53–56, match the function with its graph. [The graphs are labeled (a), (b), (c), and (d).] f(x) = - ln (-x)

In Exercises 57-60, write the exponential equation as a logarithmic equation, or vice versa. \(e^{0}=1\) Text Transcription: e^{0} = 1

In Exercises 57-60, write the exponential equation as a logarithmic equation, or vice versa. \(e^{-2}=0.1353 \ldots\) Text Transcription: e^{-2} = 0.1353 …

In Exercises 57-60, write the exponential equation as a logarithmic equation, or vice versa. \(\ln 2=0.6931 \ldots\) Text Transcription: ln 2 = 0.6931 …

In Exercises 57-60, write the exponential equation as a logarithmic equation, or vice versa. \(ln 0.5=-0.6931 \ldots\) Text Transcription: ln 0.5 = -0.6931 …

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = 3 ln x

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = -2 ln x

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = ln 2x

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = ln |x|

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = ln (x - 3)

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = ln x - 4

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = ln (x + 2)

In Exercises 61-68, sketch the graph of the function and state its domain. f(x) = ln (x - 2) + 1

In Exercises 69-72, write an equation for the function having the given characteristics. The shape of \(f(x)=e^{x}\), but shifted eight units upward and reflected in the \(x\)-axis Text Transcription: f(x) = e^x x

In Exercises 69-72, write an equation for the function having the given characteristics. The shape of \(f(x)=e^{x}\), but shifted two units to the left and six units downward Text Transcription: f(x) = e^x

In Exercises 69-72, write an equation for the function having the given characteristics. The shape of f(x) = ln x, but shifted five units to the right and one unit downward

In Exercises 69-72, write an equation for the function having the given characteristics. The shape of f(x) = ln x, but shifted three units upward and reflected in the \(y\)-axis Text Transcription: y

In Exercises 73-76, illustrate that the functions \(f\) and \(g\) are inverses of each other by using a graphing utility to graph them in the same viewing window. \(f(x)=e^{2 x}, g(x)=\ln \sqrt{x}\) Text Transcription: f g f(x) = e^{2 x}, g(x) = ln sqrt{x}

In Exercises 73-76, illustrate that the functions \(f\) and \(g\) are inverses of each other by using a graphing utility to graph them in the same viewing window. \(f(x)=e^{x / 3}, g(x)=\ln x^{3}\) Text Transcription: f g f(x) = e^{x / 3}, g(x) = ln x^{3}

In Exercises 73-76, illustrate that the functions \(f\) and \(g\) are inverses of each other by using a graphing utility to graph them in the same viewing window. \(f(x)=e^{x}-1, g(x)=\ln (x+1)\) Text Transcription: f g f(x) = e^{x}-1, g(x) = ln (x + 1)

In Exercises 73-76, illustrate that the functions \(f\) and \(g\) are inverses of each other by using a graphing utility to graph them in the same viewing window. \(f(x)=e^{x-1}, g(x)=1+\ln x\) Text Transcription: f g f(x) = e^{x-1}, g(x) = 1 + ln x

In Exercises 77-80, (a) find the inverse of the function, (b) use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window, and (c) verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\). \(f(x)=e^{4 x-1}\) Text Transcription: f f^{-1} f^{-1}(f(x)) = x f(f^ {-1}(x)) = x f(x) = e^{4x - 1}

In Exercises 77-80, (a) find the inverse of the function, (b) use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window, and (c) verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\). \(f(x)=3 e^{-x}\) Text Transcription: f f^{-1} f^{-1}(f(x)) = x f(f^ {-1}(x)) = x f(x) = 3 \e^{-x}

In Exercises 77-80, (a) find the inverse of the function, (b) use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window, and (c) verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\). f(x) = 2 ln (x-1) Text Transcription: f f^{-1} f^{-1}(f(x)) = x f(f^ {-1}(x)) = x

In Exercises 77-80, (a) find the inverse of the function, (b) use a graphing utility to graph \(f\) and \(f^{-1}\) in the same viewing window, and (c) verify that \(f^{-1}(f(x))=x\) and \(f\left(f^{-1}(x)\right)=x\). f(x) = 3 + ln (2x) Text Transcription: f f^{-1} f^{-1}(f(x)) = x f(f^ {-1}(x)) = x

In Exercises 81-86, apply the inverse properties of ln x and \(e^{x}\) to simplify the given expression. \(\ln e^{x^{2}}\) Text Transcription: e^x ln e^{x^2}

In Exercises 81-86, apply the inverse properties of ln x and \(e^{x}\) to simplify the given expression. \(\ln e^{2 x-1}\) Text Transcription: e^x ln e^{2x - 1}

In Exercises 81-86, apply the inverse properties of ln x and \(e^{x}\) to simplify the given expression. \(e^{\ln (5 x+2)}\) Text Transcription: e^x e^{ln (5x + 2)}

In Exercises 81-86, apply the inverse properties of ln x and \(e^{x}\) to simplify the given expression. \(e^{\ln \sqrt{x}}\) Text Transcription: e^x e^{ln sqrt{x}}

In Exercises 81-86, apply the inverse properties of ln x and \(e^{x}\) to simplify the given expression. \(-1+\ln e^{2 x}\) Text Transcription: e^x -1 + ln e^{2x}

In Exercises 81-86, apply the inverse properties of ln x and \(e^{x}\) to simplify the given expression. \(-8+e^{\ln x^{3}}\) Text Transcription: e^x -8 + e^{ln x^{3}}

In Exercises 87 and 88, use the properties of logarithms to approximate the indicated logarithms, given that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986\). (a) ln 6 (b) \(\ln \frac{2}{3}\) (c) ln 81 (d) \(\ln \sqrt{3}\) Text Transcription: ln 2 approx 0.6931 ln 3 approx 1.0986 ln 2 / 3 ln sqrt{3}

In Exercises 87 and 88, use the properties of logarithms to approximate the indicated logarithms, given that \(\ln 2 \approx 0.6931\) and \(\ln 3 \approx 1.0986\). (a) ln 0.25 (b) ln 24 (c) \(\ln \sqrt[3]{12}\) (d) \(\ln \frac{1}{72}\) Text Transcription: ln 2 approx 0.6931 ln 3 approx 1.0986 ln sqrt[3]{12} ln 1 / 72

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln \frac{x}{4}\) Text Transcription: ln x / 4

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(ln \sqrt{x^{5}}\) Text Transcription: ln sqrt{x^5}

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln \frac{x y}{z}\) Text Transcription: ln xy / z

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. ln (xyz)

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln \left(x \sqrt{x^{2}+5}\right)\) Text Transcription: ln (x sqrt{x^2 + 5})

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln \sqrt[3]{z+1}\) Text Transcription: ln sqrt[3]{z + 1}

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln \sqrt{\frac{x-1}{x}}\) Text Transcription: ln sqrt {x - 1 / x}

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln z(z-1)^{2}\) Text Transcription: ln z(z - 1)^{2}

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln \left(3 e^{2}\right)\) Text Transcription: ln (3e^{2})

In Exercises 89-98, use the properties of logarithms to expand the logarithmic expression. \(\ln \frac{1}{e}\) Text Transcription: ln 1 / e

In Exercises 99-106, write the expression as the logarithm of a single quantity. ln x + ln 7

In Exercises 99-106, write the expression as the logarithm of a single quantity. \(\ln y+\ln x^{2}\) Text Transcription: ln y + ln x^{2}

In Exercises 99-106, write the expression as the logarithm of a single quantity. ln (x - 2) - ln (x + 2)

In Exercises 99-106, write the expression as the logarithm of a single quantity. 3 ln x + 2 ln y - 4 ln z

In Exercises 99-106, write the expression as the logarithm of a single quantity. \(frac{1}{3}\left[2 \ln (x+3)+\ln x-\ln \left(x^{2}-1\right)\right]\) Text Transcription: 1 / 3 [2 n (x + 3) + ln x - ln (x^{2} - 1)]

In Exercises 99-106, write the expression as the logarithm of a single quantity. 2[ln x - ln (x + 1) - ln (x - 1)]

In Exercises 99-106, write the expression as the logarithm of a single quantity. \(2 \ln 3-\frac{1}{2} \ln \left(x^{2}+1\right)\) Text Transcription: 2 ln 3 - {1 / 2} ln (x^2 + 1)

In Exercises 99-106, write the expression as the logarithm of a single quantity. \(\frac{3}{2}\left[\ln \left(x^{2}+1\right)-\ln (x+1)-\ln (x-1)\right]\) Text Transcription: 3 / 2 [ln (x^2 + 1) - ln (x + 1) - ln (x - 1)]

Solving an Exponential or Logarithmic Equation In Exercises 107-110, solve for \(x\) accurate to three decimal places. (a) \(e^{\ln x}=4\) (b) \(\ln e^{2 x}=3\) Text Transcription: x e^{ln x} = 4 ln e^{2x} = 3

Solving an Exponential or Logarithmic Equation In Exercises 107-110, solve for \(x\) accurate to three decimal places. (a) \(e^{\ln 2 x}=12\) (b) \(\ln e^{-x}=0\) Text Transcription: x e^{ln 2x} = 12 ln e^{-x} = 0

Solving an Exponential or Logarithmic Equation In Exercises 107-110, solve for \(x\) accurate to three decimal places. (a) ln x = 2 (b) \(e^{x}=4\) Text Transcription: x e^{x} = 4

Solving an Exponential or Logarithmic Equation In Exercises 107-110, solve for \(x\) accurate to three decimal places. (a) \(\ln x^{2}=8\) (b) \(e^{-2 x}=5\) Text Transcription: x ln x^{2} = 8 e^{-2 x} = 5

Solving an Inequality In Exercises 111-114, solve the inequality for \(x\). \(e^{x}>5\) Text Transcription: x e^{x} > 5

Solving an Inequality In Exercises 111-114, solve the inequality for \(x\). \(e^{1-x}<6\) Text Transcription: x e^{1 - x} < 6

Solving an Inequality In Exercises 111-114, solve the inequality for \(x\). -2 < ln x < 0 Text Transcription: x

Solving an Inequality In Exercises 111-114, solve the inequality for \(x\). 1 < ln x < 100 Text Transcription: x

In Exercises 115 and 116, show that f = g by using a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Assume x > 0.) \(f(x)-\ln \frac{x^{2}}{4}\) g(x) = 2 ln x - ln 4 Text Transcription: f g f(x) - ln {x^2 / 4}

In Exercises 115 and 116, show that f = g by using a graphing utility to graph \(f\) and \(g\) in the same viewing window. (Assume x > 0.) \(f(x) =\ln \sqrt{x\left(x^{2}+1\right)}\) \(g(x) =\frac{1}{2}\left[\ln x+\ln \left(x^{2}+1\right)\right]\) Text Transcription: f g f(x) = ln sqrt{x(x^{2}+1)} g(x) = 1 / 2 [ln x + ln (x^{2} + 1)]

Stating Properties In your own words, state the properties of the natural logarithmic function.

Explain why \(\ln e^{x}=x\). Text Transcription: ln e^{x} = x

Stating Properties In your own words, state the properties of the natural exponential function.

Describe the Relationship Describe the relationship between the graphs of \(f(x)=\ln x\) and \(g(x)=e^{x}\). Text Transcription: f(x) = ln x g(x) = e^x .

The table of values below was obtained by evaluating a function. Determine which of the statements may be true and which must be false, and explain why. (a) \(y\) is an exponential function of \(x\). (b) \(y\) is a logarithmic function of \(x\). (c) \(x\) is an exponential function of \(y\). (d) \(y\) is a linear function of \(x\). Text Transcription: y x

The figure below shows the graph of \(y_{1}=\ln e^{x}\) or \(y_{2}=e^{\ln x}\). Which graph is it? What are the domains of \(y_{1}\) and \(y_{2}\)? Does \(\ln e^{x}=e^{\ln x}\) for all real values of \(x\)? Explain. Text Transcription: y_{1} = ln e^x y_{2 }= e^ln x y_{1} y_{2} ln e^{x} = e^ln x x

In Exercises 123 and 124, use the following information. The relationship between the number of decibels \(\boldsymbol{\beta}\) and the intensity of a sound in watts per centimeter squared is \(\beta=\frac{10}{\ln 10} \ln \left(\frac{I}{10^{-16}}\right)\) Use the properties of logarithms to write the formula in a simpler form. Text Transcription: beta beta = 10 / ln 10 ln (I / 10^{-16}

In Exercises 123 and 124, use the following information. The relationship between the number of decibels \(\boldsymbol{\beta}\) and the intensity of a sound in watts per centimeter squared is \(\beta=\frac{10}{\ln 10} \ln \left(\frac{I}{10^{-16}}\right)\) Determine the number of decibels of a sound with an intensity of \(10^{-5}\) watt per square centimeter. Text Transcription: beta beta = 10 / ln 10 ln (I / 10^{-16} 10^-5

In Exercises 125 and 126, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. ln (x + 25) = ln x + ln 25

In Exercises 125 and 126, determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. ln xy = ln x ln y

Use a graphing utility to graph the functions \(f(x)=6^{x}\) and \(g(x)=x^{6}\) in the same viewing window. Where do these graphs intersect? As \(x\) increases, which function grows more rapidly? Text Transcription: f(x) = 6^x g(x) = x^6 x

Use a graphing utility to graph the functions f(x) = ln x and \(g(x)=x^{1 / 4}\) in the same viewing window. Where do these graphs intersect? As \(x\) increases, which function grows more rapidly? Text Transcription: g(x) = x^{1 / 4} x

Let \(f(x)=\ln \left(x+\sqrt{x^{2}+1}\right)\). (a) Use a graphing utility to graph \(f\) and determine its domain. (b) Show that \(f\) is an odd function. (c) Find the inverse function of \(f\). Text Transcription: f(x) = ln (x + sqrt{x^2 + 1}) f

There are 25 prime numbers less than 100. The Prime Number Theorem states that the number of primes less than \(x\) approaches \(p(x) \approx \frac{x}{\ln x}\). Use this approximation to estimate the rate (in primes per 100 integers) at which the prime numbers occur when (a) x = 1000 (b) x = 1,000,000 (c) x = 1,000,000,000 Text Transcription: x p(x) approx x / ln x

For large values of \(\boldsymbol{n}\), \(n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots(n-1) \cdot n\) can be approximated by Stirling's Formula, \(n ! \approx\left(\frac{n}{e}\right)^{n} \sqrt{2 \pi n}\) In Exercises 131 and 132, find the exact value of \(n\) !, and then approximate \(n\) ! using Stirling's Formula. n = 12 Text Transcription: n n! =1 cdot 2 cdot 3 cdot 4 cdot cdots(n - 1) cdot n n! approx (n / e)^{n} sqrt{2 pi n}

For large values of \(\boldsymbol{n}\), \(n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots(n-1) \cdot n\) can be approximated by Stirling's Formula, \(n ! \approx\left(\frac{n}{e}\right)^{n} \sqrt{2 \pi n}\) In Exercises 131 and 132, find the exact value of \(n\) !, and then approximate \(n\) ! using Stirling's Formula. n = 15 Text Transcription: n n! =1 cdot 2 cdot 3 cdot 4 cdot cdots(n - 1) cdot n n! approx (n / e)^{n} sqrt{2 pi n}

Prove that ln (x / y) = ln x - ln y, x > 0, y > 0.

Prove that \(\ln x^{y}=y \ln x\). Text Transcription: ln x^y = y ln x