To solve exponential and logarithmic equations, you can

To ________ an equation in means to find all values of for which the equation is true.

To solve exponential and logarithmic equations, you can use the following One-to-One and Inverse Properties. (a) if and only if ________. (b) if and only if ________. (c) ________ (d) ________

To solve exponential and logarithmic equations, you can use the following strategies. (a) Rewrite the original equation in a form that allows the use of the ________ Properties of exponential or logarithmic functions. (b) Rewrite an exponential equation in ________ form and apply the Inverse Property of ________ functions. (c) Rewrite a logarithmic equation in ________ form and apply the Inverse Property of ________ functions.

An ________ solution does not satisfy the original equation.

(a) (b)

(a) (b)

x  1.219 x  ln 16 x  2 ln 25 x  3.7081 x  2 e x  1 ln 15 25 4e 3e x

x  ln 16 x  2 ln 25 x  3.7081 x  2 e x  1 ln 15 25 4e 3

log4 3x  3

log2 log4 3x  3 x 3  10

ln 2x 3  5.8

ln x 1  3.8

4 x  243 x  16

3 4 x  243

64 1 2 x  32

1 4 x  64

ln x ln 2  0 l

ln x ln 5  0

e x  4 x  2

e e x  4

ln x  1 l

log x  2

log4 x  3

log5 x  1 2

f x  2x

f x  27x

f x  log f x  ln x 4 3 x

f x  ln x 4 3

ex  ex22

e2x  ex28 e

ex23  ex2 e

ex2  ex22x ex

4 3   32 x   20

2 5x 4 3   32

2e x  91 x  10

4e 2e x  91 x

e x 10  47 x 9  19 4

6 e x 10  47

3 5x  3000 2x  80

6 3 5x  3000

5 3t  0.10 t 2  0.20 6

4 5 3t  0.10 t

3 x3  32 x1  27 4

2 3 x3  32 x

2 2x  431 3x  565 2

8 2 2x  431 3

e

e e 2x  50

500e 4x  75 x  300 e

1000e 500e 4x  75 x

7 2e x  11 x  5

14 3e 7 2e x  11 x

6 2  13  41 3x1  7  9 1

8 462x 6 2  13  41 3x

e x 6  0 2x 4ex 5  0 8

e2x 5e e x 6  0 2

e x 36  0 2x 3ex 4  0 e

e2x 9e e x 36  0

500 100 ex 2  20

400 1 ex  350 5

3000 2 e2x  2

119 e6x 14  7

1 0.065 365  365t  4

4 2.471 40  9t  21

1 0.10 12  12t  2

16 0.878 26  3t  30

7  2 x  212 x

5 7  2 x  212

6e x1 15  0 1x  25 5

4e 6e x1 15  0 1x

3e 2x 3  11 3x 2  962 4

8e 3e 2x 3  11 3x

e 1.8x 7  0 0.09t  3

e e 1.8x 7  0 0.

e 2.724x  29 0.125t 8  0

e e 2.724x  29

ln x  3 l

ln x  1.6

ln x 7  0 l

ln x 1  0

ln 2x  2.4

2.1  ln 6x

log x  6

log 3z  2

3ln 5x  10

2 ln x  7

lnx 2  1 ln

lnx 8  5 3

7 3 ln x  5

2 6 ln x  10

2 2 ln 3x  17

2 3 ln x  12

6 log3 0.5x  11

4 log x 6  11

ln x ln x 1  2

ln x ln x 1  1

ln x ln x 2  1 l

ln x ln x 3  1

ln x 5  ln x 1 ln x 1 ln

ln x 1 ln x 2  ln x l

log2 2x 3  log2 x 4

log 3x 4  log x 10

log x 4 log x  log x 2

log2 x log2 x 2  log2 x 6

log4 x log4 x 1  1 2 l

log3 x log3 x 8  2

log 8x log 1 x   2 l

log 4x log 12 x   2 l

3 ln x  0 1

10 4 ln x 2  0 l

2 ln x 3  3 l

ln x 1  2 ln x

r  0.05

r  0.045

r  0.025

r  0.0375

2x x  0 2e2x 2xe2x  0

x2ex 2xe 2x x  0 2e2

xe 2x  0 x ex  0 x2ex

e2x 2xe xe 2x  0 x

2x ln x x  0

1 ln x x 2x ln x x  0 2  0

1 ln x 2  0

2x ln 1 x x  0 1

DEMAND The demand equation for a limited edition coin set is Find the demand for a price of

DEMAND The demand equation for a hand-held electronic organizer is Find the demand for a price of (a) and (b) p  $400.

FOREST YIELD The yield (in millions of cubic feet per acre) for a forest at age years is given by (a) Use a graphing utility to graph the function. (b) Determine the horizontal asymptote of the function. Interpret its meaning in the context of the problem. (c) Find the time necessary to obtain a yield of 1.3 million cubic feet.

TREES PER ACRE The number of trees of a given species per acre is approximated by the model where is the average diameter of the trees (in inches) 3 feet above the ground. Use the model to approximate the average diameter of the trees in a test plot when

U.S. CURRENCY The values (in billions of dollars) of U.S. currency in circulation in the years 2000 through 2007 can be modeled by where represents the year, with corresponding to 2000. During which year did the value of U.S. currency in circulation exceed $690 billion? (

MEDICINE The numbers of freestanding ambulatory care surgery centers in the United States from 2000 through 2007 can be modeled by where represents the year, with corresponding to 2000. (Source: Verispan) (a) Use a graphing utility to graph the model. (b) Use the trace feature of the graphing utility to estimate the year in which the number of surgery centers exceeded 3600.

AVERAGE HEIGHTS The percent of American males between the ages of 18 and 24 who are no more than inches tall is modeled by and the percent of American females between the ages of 18 and 24 who are no more than inches tall is modeled by (Source: U.S. National Center for Health Statistics) (a) Use the graph to determine any horizontal asymptotes of the graphs of the functions. Interpret the meaning in the context of the problem. (b) What is the average height of each sex?

LEARNING CURVE In a group project in learning theory, a mathematical model for the proportion of correct responses after trials was found to be (a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. (c) After how many trials will 60% of the responses be correct?

AUTOMOBILES Automobiles are designed with crumple zones that help protect their occupants in crashes. The crumple zones allow the occupants to move short distances when the automobiles come to abrupt stops. The greater the distance moved, the fewer gs the crash victims experience. (One g is equal to the acceleration due to gravity. For very short periods of time, humans have withstood as much as 40 gs.) In crash tests with vehicles moving at 90 kilometers per hour, analysts measured the numbers of gs experienced during deceleration by crash dummies that were permitted to move meters during impact. The data are shown in the table. A model for the data is given by where is the number of gs.

DATA ANALYSIS An object at a temperature of was removed from a furnace and placed in a room at The temperature of the object was measured each hour and recorded in the table. A model for the data is given by The graph of this model is shown in the figure.

The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.

The logarithm of the sum of two numbers is equal to the product of the logarithms of the numbers.

The logarithm of the difference of two numbers is equal to the difference of the logarithms of the numbers.

The logarithm of the quotient of two numbers is equal to the difference of the logarithms of the numbers.

THINK ABOUT IT Is it possible for a logarithmic equation to have more than one extraneous solution? Explain.

FINANCE You are investing dollars at an annual interest rate of compounded continuously, for years. Which of the following would result in the highest value of the investment? Explain your reasoning. (a) Double the amount you invest. (b) Double your interest rate. (c) Double the number of years.

THINK ABOUT IT Are the times required for the investments in Exercises 117120 to quadruple twice as long as the times for them to double? Give a reason for your answer and verify your answer algebraically.

The effective yield of a savings plan is the percent increase in the balance after 1 year. Find the effective yield for each savings plan when $1000 is deposited in a savings account. Which savings plan has the greatest effective yield? Which savings plan will have the highest balance after 5 years? (a) 7% annual interest rate, compounded annually (b) 7% annual interest rate, compounded continuously (c) 7% annual interest rate, compounded quarterly (d) 7.25% annual interest rate, compounded quarterly

GRAPHICAL ANALYSIS Let and where (a) Let and use a graphing utility to graph the two functions in the same viewing window. What do you observe? Approximate any points of intersection of the two graphs. (b) Determine the value(s) of for which the two graphs have one point of intersection. (c) Determine the value(s) of for which the two graphs have two points of intersection.

CAPSTONE W