Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Aim: How do we solve exponential equations using logarithms?

Do Now:

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Solving Exponential Equations

What is an exponential equation?

An exponential equation is an equation, in which the variable is an exponent. Ex. 3x - 4 = 9

Question: What power of 3 will give us 9?Answer: 2 x - 4 = 2 and x = 6

Solution of this equation was possible because 9 is a power of 3 or in general terms

For b > 0, and b 1, bx = by x = y

To solve an exponential equation, write eachside as a power of the same base.

3x - 4 = 32

x - 4 = 2rewrite each side with same base

equate the exponents and solve

x = 6

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Exponential Equation Problems

Solve and check 5x + 1 = 54

Solve and check 2x - 1 = 82

x + 1 = 4 bx = by x = y5x + 1 = 54

x = 3

Check 53 + 1 = 54 54 = 54

2x - 1 = 82 convert to like bases8 = 23

2x - 1 = (23)2 = 26

x – 1 = 6 bx = by x = yx = 7

Check 27 - 1 = 82 26 = 64 = 82

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Exponential Equation Problems

Solve and check 9x + 1 = 27x

9x + 1 = 27x

(32)x + 1 = (33)x

2x + 2 = 3x

2 = x

convert to like bases9 = 32; 27 = 33

distributive property;product of powers property32x + 2 = 33x

bx = by x = y

Check 92 + 1 = 272

93 = 272

729 = 729

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Exponential Equation Problems

Solve and check

(1

4)x 81 x

convert to like bases1/4 = 2-2; 8 = 23

distributive property;product of powers property

bx = by x = y

(2 2)x (23)1 x

(1

4)x 81 x

22x23 3x

-2x = 3 – 3x

x = 3

Check

(1

4)3 81 3 8 2

1

64

1

82 1

64

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Solving Exponential Equations w/Logs

What if the two sides of the exponentialequation cannot be expressed with the samebase? Ex. 9x = 14

1. Write the log of each side log 9x = log 14

2. Use the power rule to simplify x log 9 = log 14

3. Solve for x

x log14

log9

4. Evaluate on calculator

x 1.1461280.9542425

1.202

5. Check 91.202 = 14

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Alternate Method - 1

Solve for x to the nearest 10th:12 • 12x = 500

Method 1

log (12 • 12x) = log 500

log 12 + log 12x = log 500

log 12 + x log 12 = log 500

x log 12 = log 500 - log 12

x log500 log 12

log12

x = 1.5 to nearest 10th

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Alternate Method - 2

Solve for x to the nearest 10th:12 • 12x = 500

Method 2

12x 500

12

log12x log(500

12)

x log12 log 500 log12

x log500 log 12

log12= 1.5

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Alternate Method - 3

Solve for x to the nearest 10th:12 • 12x = 500

Method 3

121 + x = 500

log 121 + x = log 500

(1 + x)log 12 = log 500

1 x log500

log12

x log500

log12 1 1.5

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Problems

Solve 6x = 42 to 3 decimal places

x log 42

log 6

x 1.62324929

0.77815125042.086

log 6x = log 42 Property of Equality for Log functions

x log 6 = log 42Power Property of Logarithms

Solve 3.1a – 3 = 9.42 to 3 decimal places

Solve 9a = 2a a = 4.982a = 0

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Complicated Problem

Solve 82x - 5 = 5x + 1

2x log 8 - x log 5 = log 5 + 5 log 8

x log5 5log8

2log8 log5

x 0.6990 5(0.9031)

2(0.9031) 0.69904.7095

log 82x - 5 = log 5x + 1 Property of Equality for Log functions

(2x - 5)log 8 = (x + 1)log 5Power Property of Logarithms

2x log 8 - 5 log 8 = x log 5 + 1 log 5Distributive Property

x(2 log 8 - log 5) = log 5 + 5 log 8Distributive Property

Aim: Exponential Equations using Logs

Course: Alg. 2 & Trig.

Problems

Solve 4e2x = 5 to 3 decimal places

x 1

2ln

5

4

x 1

2.2231435513 0.112

ln e2x = ln 5/4 Property of Equality for Ln functions

2x = ln 5/4Inverse Property of Logs & Expos

e2x = 5/4 Divide both sides by 4

Check: 4e2(0.112) = 5