Solving Exponential and Solving Exponential and Logarithmic EquationsLogarithmic Equations

Section 3.4Section 3.4

JMerrill, 2005JMerrill, 2005

Revised, 2008Revised, 2008

Same BaseSame Base

Solve: 4Solve: 4x-2 x-2 = 64= 64xx

44x-2x-2 = (4 = (433))xx

44x-2x-2 = 4 = 43x3x

x–2 = 3xx–2 = 3x -2 = 2x-2 = 2x -1 = x-1 = x

If bM = bN, then M = N64 = 43

If the bases are already =, just solve the exponents

You DoYou Do

Solve 27Solve 27x+3x+3 = 9 = 9x-1x-1

x 3 x 13 2

3x 9 2x 2

3 3

3 3

3x 9 2x 2

x 9 2

x 11

Review – Change Logs to Review – Change Logs to ExponentsExponents

loglog33x = 2x = 2

loglogxx16 = 216 = 2

log 1000 = xlog 1000 = x

32 = x, x = 9

x2 = 16, x = 4

10x = 1000, x = 3

Using Properties to Solve Using Properties to Solve Logarithmic EquationsLogarithmic Equations

If the exponent is a variable, then take the If the exponent is a variable, then take the natural log of both sides of the equation and natural log of both sides of the equation and use the appropriate property. use the appropriate property.

Then solve for the variable.Then solve for the variable.

Example: SolvingExample: Solving

22x x = 7= 7 problemproblem ln2ln2xx = ln7 = ln7 take ln both sidestake ln both sides xln2 = ln7xln2 = ln7 power rulepower rule x = x = divide to solve for divide to solve for

xx

x = 2.807x = 2.807

ln7ln2

Example: SolvingExample: Solving

eex x = 72= 72 problemproblem lnelnexx = ln 72 = ln 72 take ln both sidestake ln both sides x lne = ln 72x lne = ln 72 power rulepower rule x = 4.277x = 4.277 solution: becausesolution: because

ln e = ?ln e = ?

You Do: SolvingYou Do: Solving

2e2ex x + 8 = 20+ 8 = 20 problemproblem 2e2exx = 12 = 12 subtract 8subtract 8 eexx = 6 = 6 divide by 2divide by 2 ln eln exx = ln 6 = ln 6 take ln both sidestake ln both sides x lne = 1.792x lne = 1.792 power rulepower rule

x = 1.792x = 1.792 (remember: lne = (remember: lne = 1)1)

ExampleExample

Solve 5Solve 5x-2x-2 = 4 = 42x+32x+3

ln5ln5x-2x-2 = ln4 = ln42x+32x+3

(x-2)ln5 = (2x+3)ln4(x-2)ln5 = (2x+3)ln4 The book wants you to distribute…The book wants you to distribute… Instead, divide by ln4Instead, divide by ln4 (x-2)1.1609 = 2x+3(x-2)1.1609 = 2x+3 1.1609x-2.3219 = 2x+31.1609x-2.3219 = 2x+3 x≈6.3424x≈6.3424

Solving by Rewriting as an Solving by Rewriting as an ExponentialExponential

Solve logSolve log44(x+3) = 2(x+3) = 2

4422 = x+3 = x+3 16 = x+316 = x+3 13 = x13 = x

You DoYou Do

Solve 3ln(2x) = 12Solve 3ln(2x) = 12 ln(2x) = 4ln(2x) = 4 Realize that our base is e, soRealize that our base is e, so ee44 = 2x = 2x x x ≈ 27.299≈ 27.299

You always need to check your answers You always need to check your answers because sometimes they don’t work!because sometimes they don’t work!

Using Properties to Solve Using Properties to Solve Logarithmic EquationsLogarithmic Equations

1.1. Condense both sides first (if Condense both sides first (if necessary).necessary).

2.2. If the bases are the same on both If the bases are the same on both sides, you can cancel the logs on both sides, you can cancel the logs on both sides.sides.

3.3. Solve the simple equationSolve the simple equation

Example: Solve for xExample: Solve for x

loglog336 = log6 = log333 + log3 + log33xx problemproblem

loglog336 = log6 = log333x3x condensecondense

6 = 3x6 = 3x drop logsdrop logs

2 = x2 = x solutionsolution

You Do: Solve for xYou Do: Solve for x

log 16 = x log 2 log 16 = x log 2 problemproblem

log 16 = log 2log 16 = log 2xx condensecondense

16 = 2x16 = 2x drop logsdrop logs

x = 4 x = 4 solutionsolution

You Do: Solve for xYou Do: Solve for x

loglog44x = logx = log4444 problemproblem

= log= log4444 condensecondense

= 4= 4 drop logsdrop logs

cube each sidecube each side

X = 64X = 64 solutionsolution

13

13

4log x

13x

3133 4x

ExampleExample

7xlog7xlog225 = 3xlog5 = 3xlog225 + ½ log5 + ½ log222525

loglog22557x7x = log = log22553x3x + log + log2225 25 ½ ½

loglog22557x7x = log = log22553x3x + log + log225511

7x = 3x + 17x = 3x + 1 4x = 14x = 1

14

x

You DoYou Do

Solve:Solve: loglog777 + log7 + log772 = log2 = log77x + logx + log77(5x – 3) (5x – 3)

You Do AnswerYou Do Answer

Solve:Solve: loglog777 + log7 + log772 = log2 = log77x + logx + log77(5x – 3)(5x – 3)

loglog7714 = log14 = log7 7 x(5x – 3)x(5x – 3)

14 = 5x14 = 5x22 -3x -3x 0 = 5x0 = 5x22 – 3x – 14 – 3x – 14 0 = (5x + 7)(x – 2)0 = (5x + 7)(x – 2) 7

,25

x

Do both answers work? NO!!

Final ExampleFinal Example

How long will it take for $25,000 to grow to How long will it take for $25,000 to grow to $500,000 at 9% annual interest $500,000 at 9% annual interest compounded monthly?compounded monthly?

0( ) 1

ntrA t A

n

ExampleExample

0( ) 1

ntrA t A

n12

0.09500,000 25,000 1

12

t

1220 1.0075 t

12t ln(1.0075) ln20

ln20t

12ln1.0075t 33.4