Chapter 10 INVERSE, EXPONENTIAL, AND LOGARITHMIC
FUNCTIONS
10.7 Exponential and Logarithmic Equations and Their Applications Learning Objectives
1 Solve equations involving variables in the exponents.
2 Solve equations involving logarithms.
3 Solve applications involving compound interest.
4 Solve applications involving base e exponential growth and decay.
5 Use the change-of-base rule.
Key Terms
Use the vocabulary terms listed below to complete each statement in exercises 1−2.
compound interest continuous compounding
1. The formula for ________________________ is A Pe rt.
2. The formula for ________________________ is A P
1nr nt. Objective 1 Solve equations involving variables in the exponents. Video ExamplesReview these examples for Objective 1:
1. Solve 4x=30. Approximate the solution to three decimal places.
4 30
If , and 0, 0, log 4 log 30 then log log .
log 4 log 30 Power rule log 30 Divide by log 4. log 4 2.453 Use a calculator. b b x x y x y x x y x x x = = > > = = = = » Now Try:
1. Solve 3x=15. Approximate the solution to three decimal places. _____________
2. Solve e0.005x=9.Approximate the solution to three decimal places.
0.005 0.005 9 If , and 0, ln ln 9 0, then ln ln . 0.005 ln ln 9 Power rule 0.005 ln 9 ln 1 ln 9 Divide by 0.005. 0.005 439.445 Use a calculator. x x e x y x e y x y x e x e x x = = > = > = = = = = »
The solution set is {439.445}.
2. Solve e0.4x=15. Approximate the solution to three decimal places.
_____________
Objective 1 Practice Exercises
For extra help, see Examples 1–2 on page 692 of your text.
Solve each equation. Give solutions to three decimal places.
1. 25x2 1253x 1. ________________
2. 4x132x 2. ________________
Objective 2 Solve equations involving logarithms. Video Examples
Review these examples for Objective 2:
3. Solve log3(x -1)2= Give the exact solution. 3. ( ) ( ) ( ) 2 3 2 3 2 log 1 3
1 3 Write in exponential form. 1 27
Take the square 1 27 root on each side.
1 3 3 Simplify the square root. 1 3 3 Add 1. x x x x x x - = - = - = - = - = =
Since the domain of logbx is
(
0, ,¥)
we disregard the negative solution, 1 3 3. -Check: ( )(
)
(
)
2 3 ? 2 3 ? 2 3 ? 3 ? 3 log 1 3 log 1 3 3 1 3 log 3 3 3 log 27 3 3 27 27 27 x- = + - = = = = =The solution set is
{
1 3 3 .+}
Now Try:
3. Solve log6(x +1)3= Give 2. the exact solution.
_____________
4. Solve log 53( x+42)-log3x=log 26.3
( )
3 3 3
3 3
log 5 42 log log 26. 5 42 log log 26 If log log 5 42 26 b b x x x x x y x + - = + = = + = 4. Solve ( ) 6 6 6
log 2x+ -7 log x=log 16. _____________
Check: ( ) ( ) 3 3 3 ? 3 3 3 ? 3 3 3 ? 52 3 2 3 3 3
log 5 42 log log 26 log 5 2 42 log 2 log 26 log 52 log 2 log 26 log log 26 log 26 log 26 x+ - x= ⋅ + - = - = = =
The solution set is {2}.
5. Solve log2(x+ +7) log2(x+ =3) log 77.2
( ) ( ) ( )( )
[
]
( )( ) ( )( ) 2 2 2 2 2 2 2log 7 log 3 log 77
log 7 3 log 77 Product rule 7 3 77 If log log then . 10 21 77 Multiply. 10 56 0 Subtract 77. 4 14 0 Factor. b b x x x x x x x y x y x x x x x x + + + = + + = + + = = = + + = + - = - + = 4 0 or 14 0 4 14 x x x x - = + = = =
-The value −14 must be rejected since it leads to the logarithm of a negative number in the original equation.
A check shows that the only solution is 4. The solution set is {4}.
5. Solve.
( ) ( )
4 4 4
log 4x- +3 log x=log 2x-1 _____________
Objective 2 Practice Exercises
For extra help, see Examples 3–5 on pages 693–694 of your text.
Solve each equation. Give exact solution.
5. log (3 x210) log 3x 1 5. ________________
6. ln(x 4) ln(x 2) ln 7 6. ________________
Objective 3 Solve applications involving compound interest. Video Examples
Review these examples for Objective 3:
6. How much money will there be in an account at the end of 5 years if $5000 is deposited at 4% compounded monthly?
Because interest is compounded monthly,
n = 12. The other given values are P = 5000, r = 0.04, and t = 5.
( )
(
)
( ) 12 5 60 1 0.04 5000 1 12 5000 1.0033 nt r A P n A A ⋅ = + = + = Now Try:6. How much money will there be in an account at the end of 5 years if $10,000 is deposited at 4% compounded quarterly? _____________
7. Approximate the time it would take for money deposited in an account paying 5% interest compounded quarterly to double. Round to the nearest hundredth.
We want the number of years t for P dollars to grow to 2P dollars at a rate of 5% per year. In the compound interest formula, we substitute 2P for A, and let r = 0.05 and n = 4.
(
)
4 4 4 0.05 2 1 4 2 1.0125 log 2 log1.025 log 2 4 log1.0125 log 2 4log1.0125 13.95 t t t P P t t t = + = = = = »It will take about 13.95 years for the investment to double.
7. Approximate the time it would take for money deposited in an account paying 5% interest compounded monthly to double. Round to the nearest hundredth. _____________
8. Suppose that $5000 is invested at 4% interest for
3 years. 8. Suppose that $5000 is invested at 2% interest for 3 years.
a. How much will the investment be worth if it is compounded continuously? 0.04 3 0.12 5000 5000 5637.48 rt A Pe A e A e A ⋅ = = = =
The investment will be worth $5637.48.
a. How much will the investment be worth if it is compounded continuously? _____________
b. Approximate the amount of time it would take for the investment to double. Round to the nearest tenth.
Find the value of t that will cause A to be 2($5000) = $10,000.
b. Approximate the amount of time it would take for the investment to double. Round to the nearest tenth.
0.04 0.04 0.04 10,000 5000 2 Divide by 5000. If , ln 2 ln then ln ln . ln 2 0.04 ln ln 2 Divide by 0.04. 0.04 17.3 rt t t t k A Pe e e x y e x y t e k t t = = = = = = = = = »
It will take about 17.3 years for the amount to double.
Objective 3 Practice Exercises
For extra help, see Examples 6–8 on pages 695–696 of your text.
Solve each problem.
7. How much will be in an account after 10 years if $25,000 is invested at 8% compounded quarterly? Round to the nearest cent.
7. ________________
8. How much will be in an account after 5 years if $10,000 is invested at 4.5% compounded continuously? Round to the nearest cent.
8. ________________
9. How long will it take an investment to double if it is
Objective 4 Solve applications involving base e exponential growth and decay. Video Examples
Review these examples for Objective 4:
9. A sample of 500 g of lead-210 decays according to the function y=y e0 -0.032t, where t is the time in years, y is the amount of the sample at time t, and y is the initial amount present 0
at t = 0.
Now Try:
9. Cesium-137, a radioactive isotope used in radiation therapy, decays according to the function y=y e0 -0.0231t, where t is the time in years and
0
y is the initial amount present
at t = 0.
a. How much lead will be left in the sample after 20 years? Round to the nearest tenth of a gram.
Let t = 20 and y0=500.
0.032 20
500 263.6
y= e- ⋅ »
There will be about 263.6 grams after 20 years.
a. If an initial sample contains 36 mg of cesium-137, how much cesium-137 will be left in the sample after 50 years? Round to the nearest tenth. _____________
b. Approximate the half-life of lead-210 to the nearest tenth. Let 1 500( ) 250. 2 y = = 0.032 0.032 0.032 250 500 0.5 ln 0.5 ln ln 0.5 0.032 ln 0.5 21.7 0.032 t t t e e e t t -= = = = -= »
-The half-life of lead-210 is about 21.7 years.
b. Approximate the half-life of cesium-137 to the nearest tenth. _____________
Objective 4 Practice Exercises
For extra help, see Example 9 on pages 696–697 of your text.
Solve each problem.
10. Radioactive strontium decays according to the function y=y e0 -0.0239t, where t is the time in years. If an initial sample contains y0=15 g of radioactive strontium, how many grams will be present after 25 years? Round to the nearest hundredth of a gram.
10. ________________
11. How long will it take the initial sample of strontium in exercise 19 to decay to half of its original
amount?
11. ________________
12. The concentration of a drug in a person’s system decreases according to the function C t( )=2e-0.2t,
where C(t) is given in mg and t is in hours. How much of the drug will be in the person’s system after one hour? Approximate answer to the nearest
hundredth.
Objective 5 Use the change-of-base rule. Video Examples
Review this example for Objective 5: 10. Evaluate log 28 to four decimal places. 7
7 log 28
log 28 1.7124
log 7
= =
Now Try:
10. Evaluate log 180 to four 5 decimal places.
_____________
Objective 5 Practice Exercises
For extra help, see Example 10 on page 697 of your text.
Use the change-of-base rule to find each logarithm. Give approximations to four decimal places.
13. log 27 16 13. _________________
14. log 0.25 6 14. _________________