5-4 Exponential & 5-4 Exponential & Logarithmic EquationsLogarithmic Equations

Strategies and PracticeStrategies and Practice

ObjectivesObjectives– Use like bases to solve exponential

equations.– Use logarithms to solve exponential

equations.– Use the definition of a logarithm to solve logarithmic equations.– Use the one-to-one property of logarithms to

solve logarithmic equations.

Use like bases to solve Use like bases to solve exponential equationsexponential equations

• Equal bases must have equal exponents

EX: Given 3x-1 = 32x + 1 then x-1 = 2x+1 x = -2

If possible, rewrite to make bases equal EX: Given 2-x = 4x+1 rewrite 4 as 22

2-x = 22x+2 then –x=2x+2 x=-2/3

Note: Isolate function if needed 3(2x)=48 2x =16

You try… 1. 4x = 83

2. 5x-2 = 25x

3. 6(3x+1) = 54

4. e–x2 = e-3x - 4

(22)x = (23)3 (2)2x = (2)9

So 2x = 9, or x = 4.5

5x-2 = (52)x 5x-2 = (5)2x So x - 2 = 2x, or x = -

2 3x+1 = 9 3x+1 = 32

So x + 1= 2, or x = 1

So –x2 = -3x - 4 & x2 – 3x – 4 = 0 & (x-4)(x+1) = 0 & x=4 and x = -1

Exponentials of Unequal BasesExponentials of Unequal Bases• Use logarithm (inverse function) of same base

on both sides of equation

Solve: ex = 72

ex = 72

loge logex = ln 72 4.277

Solve: 7x-1 = 12 7x-1 = 12log7 log7

x - 1 = log7 12x = log7 12 + 1x = + 1

log 12log 7

2.277

You try…You try…1. Solve 3(2x) = 42

2. Solve 32t-5 = 15

3. Solve e2x = 5

4. Solve ex + 5 = 60

x = log2 14 3.807

t = 1/2(log3 15 + 5) 3.732

x = 1/2 ln 5 0.805

x = ln 55 4.007

Solving Logarithmic EquationsSolving Logarithmic Equations• Rewrite into exponential form

EX: Solve: ln x = - 1/2

loge x = - 1/2

e -1/2 = xx = e -1/2

EX: Solve: 2 log5 3x = 4 log5 3x = 2

52= 3x 25= 3x

25/3= x x = 25/3

0.607

8.333

Solving Logarithmic EquationsSolving Logarithmic Equations• Use properties of logarithms to condense.

EX: Solve: log4x + log4(x-1) = ½log4 x(x – 1) = 1/2

4 1/2 = x(x – 1)2 = x2 – x0 = x2 – x – 20 = (x – 2)(x + 1)x = 2 & x = -1

Check for extraneous roots.

You try…You try…1. Solve ln x = -7

2. Solve 2 log3 2x = 4

3. Solve ln x + ln (x-3) = 0

4. Solve 5 + 2ln x = 4

x = e-7 0.000912

x = 9/2

x = & x = 3 + 132

3 - 132

x = e-1/2 0.607

Double-Sided Log Equations• Equate powers (domain solutions only)

EX: Solve: log5(5x-1) = log5(x+7) 5x – 1 = x +

7 4x = 8x = 2

EX: Solve: ln(x-2) + ln(2x-3) = 2lnx

Use the properties to condense.

ln (x-2)(2x-3) = ln x2

(x-2)(2x-3) = x2

2x2 – 7x + 6 = x2

x2 – 7x + 6 = 0 (x – 6)(x – 1)= 0

x = 6 & x = 1

Check for extraneous roots.

x = -2 + 22 & x = -2 - 2 2

You try…

1. Solve ln3x2 = lnx

2. Solve log6(3x + 14) – log6 5 = log6 2x

3. Solve log2x+log2(x+5) = log2(x+4)

x = 0 & x = 1/3

x = 2

SUMMARY• Equal bases Equal exponents

• Unequal bases Apply log of given base

• Single side logs Convert to exp form

• Double-sided logs Equate powers

Note: Any solutions that result in a log(neg) cannot be used!