8-5 Exponential & Logarithmic Equations

8-5 Exponential & 8-5 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

If possible, rewrite to make bases equal

EX: Given 2-x = 4x+1

Isolate function if needed—

3(2x)=48

You try…

1. 4x = 83

2. 5x-2 = 25x

3. 6(3x+1) = 54

4. e–x2 = e-3x - 4

Solving Logarithmic EquationsSolving Logarithmic Equations• Convert to exponential (inverse) form

EX: Solve: 2log53x = 4

Now you try….Now you try….

• log x = 6

• log 5x = 3

Solving Logarithmic EquationsSolving Logarithmic EquationsUse Properties of Logs to condense

EX: Solve: log4x + log4(x-1) = ½

You try…You try…

Solve lnx+ln(x-3) = 1

Solve log x + log (x + 2) = log (x + 6)

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

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

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

You try…You try…

1. Solve ln3x2 = lnx 2. Solve log6(3x + 14) – log6 5 = log6 2x

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

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

base on both sides of equation

EX: Solve: ex = 72

Now you try….Now you try….

• 2ex + 2 = 12

• ex – 9 = 19

• 7 - 2ex = 5

• EX: Solve: 7x-1 = 12

You try…You try…

1. Solve 3(2x) = 42

2. Solve 32t-5 = 15

3. Solve e2x = 5

4. Solve ex + 5 = 60

Solving Logarithmic EquationsSolving Logarithmic Equations• Convert to exponential (inverse) form EX: Solve: lnx = -1/2

Now you tryNow you try

• ln (2x – 1) = 0

• ln x = -3

• 3ln 5x = 10

Solve lnx+ln(x-3) = 1

SUMMARYSUMMARY• 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!