MAT 150 – Class #19

Objectives

Solve an exponential equation by writing it in logarithmic form

Convert logarithms using the change of base formulaSolve an exponential equation by using properties of logarithms

Solve logarithmic equationsSolve exponential and logarithmic inequalities

Solving Exponential Equations Using Logarithmic FormsTo solve an exponential equation using logarithmic form:1. Rewrite the equation with the term containing the

exponent by itself on one side.2. Divide both sides by the coefficient of the term

containing the exponent.3. Change the new equation to logarithmic form.4. Solve for the variable.

Example

Solve the equation for t by converting it to logarithmic form and graphically to confirm the solution.

SolutionDivide both sides of the equation by 150.

Rewrite in logarithmic form.Solve for t.

43000 150(10 )t

43000 150(10 )t

420 10 t

4 log20t

log20

0.325264

t

Example (cont)

Solve the equation for t by converting it to logarithmic form and graphically to confirm the solution.

SolutionTo solve graphically enter 3000 for y1 and for y2

43000 150(10 )t

4150(10 )t

Example

a. Prove that the time it takes for an investment to double its value is if the interest rate is r, compounded continuously.

Solutiona.

ln2t

r

rtS Pe

ln2t

r

Example (cont)

b. Suppose $2500 is invested in an account earning 6% annual interest, compounded continuously. How long will it take for the amount to grow to $5000?

Solution

b. ln2t

r

ln211.5525

0.06t

Change of Base

We can use a special formula called the change of base formula to rewrite logarithms so that the base is 10 or e. The general change of base formula is summarized below.

Example

Evaluate

Solution

8log 124.

8log 124 log124

2.318 approximatelylog8

Example (cont)

b. Graph the function by changing each logarithm to a common logarithm and then by changing the logarithm to a natural logarithm.

Solutionchange to base 10

change to base e

3logy x

3logy x

log

log3

x

Example

If $10,000 is invested for t years at 10%, compounded annually, the future value is given by

In how many years will the investment grow to $45,950?Solution

10,000(1.10 )tS

45,950 10,000(1.10 )t

4.5950 1.10t

1.10log 4.5950t

1.10

log4.5950log 4.5950 16

log1.10t

The investment will grow to $45,950 in 16 years.

SOLVING EXPONENTIAL EQUATIONS USING LOGARITHMIC PROPERTIESTo solve an exponential equation using logarithmic properties:1. Rewrite the equation with a base raised to a power on

one side.2. Take the logarithm, base e or 10, of both sides of the

equation.3. Use a logarithmic property to remove the variable

from the exponent.4. Solve for the variable. 

Example

Solve the following exponential equations.a. b.

Solutiona. Take log of base 10 of both sides.

Using the Power Property of Logarithms

Solving for x

24096 8 x 3 26(4 ) 120x

24096 8 x

2log4096 log8 x

log4096 2 log8x

log4096

2log8

2

x

x

Example (cont)

Solution3 2b. 6(4 ) 120x

3 26(4 ) 120

6 6

x

3 24 20x

3 2ln4 ln20x

(3 2)ln4 ln20x

ln20

3 2ln4

x

1 ln202

3 ln4

1.387

x

x

Example

Solve by converting to exponential form and verify the solution graphically.SolutionDivide both sides by 4:

Write in exponential form:

34log 8x

34log 8x

3log 2x

23

1

9

x

x

Example

Solve by converting to exponential form and verify the solution graphically.Solution

6 3ln 12x

6 3ln 12x

3ln 6x

ln 2x

2x e

Example

Solve by converting to exponential form and then using algebraic methods.Solution

ln 3 ln( 4)x x

ln 3 ln( 4)x x

3 ln( 4) lnx x

43 ln

x

x

3 4xe

x

3 4e x x

3 4e x x

3( 1) 4x e

3

40.21

1x

e

Example

After the end of an advertising campaign, the daily sales of Genapet fell rapidly, with daily sales given byS = 3200e-0.08x dollars, where x is the number of days from the end of the campaign. For how many days after the campaign ended were sales at least $1980?Solution

0.083200 1980xe 0.08 0.61875xe 0.08ln ln0.61875xe

0.08 0.4801x6x

Assignment

Pg. 350-354#3-7 odd#15-33 odd#47#66