7.6 – Solve Exponential and Log Equations

Property of Equality for Exponential Equations

If b is a positive number other than 1,

then ___________ if and only if

___________.

x yb b

x y

Solve the equation.

3 4 57 7x 1.

3x – 4 = 5

3x = 9

x = 3

Solve the equation.

2.

3x – 8 = 4(13 – 3x)

15x = 60

x = 4

3 8 13 33 81x x 3 8 4(13 3 )3 3x x

3x – 8 = 52 – 12x

15x – 8 = 52

Solve the equation.

3.

3(4x – 1) = 2(3x + 8)

6x = 19

3(4 1) 2(3 8)3 3x x

12x – 3 = 6x + 16

6x – 3 = 16

4 1 3 827 9x x

19

6x

31

42

xx

Solve the equation.

4.

2x = –(x – 3)

x = 1

2 ( 3)2 2x x

2x = –x + 3

3x = 3

How to solve for a power

To eliminate the power, take the log of both sides. It lowers the exponent.

xa blog logxa b

log logx a blog

log

bx

a

Solve the equation.

5. 2 5x

log 2 log5x

log 2 log5x

log5

log 2x 2.322

Solve the equation.

6.

9log 7 log15x

9 log 7 log15x

log15

9 log 7x 0.155

97 15x

Solve the equation.

7.

log5 log 65x

log5 log 65x

log 65

log5x 2.594

5 25 40x 5 65x

Solve the equation.

8.

0.3ln ln 5xe

0.3 ln ln 5x e

ln 5

0.3 lnx

e

5.365

0.34 20xe

0.34 7 13xe

0.3 5xe

Property of Equality for Logarithmic Equations

If b is a positive number

other than 1, then

__________________ if and

only if ___________.

log logb bx y

x y

Solve the equation. Check for extraneous solutions.

5 5log 5 9 log 6x x 9.

5x + 9 = 6x

9 = x

5 5log 54 log 54

5x + 18 = 7x – 8

x = 13

18 = 2x – 8

26 = 2x

Solve the equation. Check for extraneous solutions.

10. ln 5 18 ln 7 8x x ln 83 ln 83

12x – 11 = 3x + 16

x = 3

9x – 11 = 16

9x = 27

Solve the equation. Check for extraneous solutions.

11. log 12 11 log 3 16x x log 25 ln 25

3x – 10 = 14 – 5x

x = 3

8x – 10 = 14

8x = 24

Solve the equation. Check for extraneous solutions.

12. log 1 log 1 6 6log 3 10 log 14 5x x

No solution

How to solve for a Log

To eliminate the log, raise both sides to the base power. This eliminates the log.

logb x a

ax b

b b

Solve the equation. Check for extraneous solutions.

13. 2log 6 5x 2 2

x – 6 = 25

x – 6 = 32

x = 38

2log 32 5

25 = 32

Solve the equation. Check for extraneous solutions.

14.

4 4

8x = 42

8x = 16

43log 8 6x

4log 8 2x

x = 2

43log 16 6

42 = 16

4log 16 2

4 4log 12 log 3x x

Solve the equation. Check for extraneous solutions.

15.

4 4x(x + 12) = 43

4log 12 3x x

4log 64 3

43 = 64

x2 + 12x = 64

x2 + 12x – 64 = 0

(x + 16)(x – 4) = 0x

x

16

-4 x = 4, -16

Solve the equation. Check for extraneous solutions.

16.

10 105x(x – 1) = 102

log5 1 2x x

log100 2

102 = 100

log5 log 1 2x x

5x2 – 5x = 100

5x2 – 5x – 100 = 0

5(x2 – x – 20) = 05(x – 5)(x + 4) = 0

x

x

-5

4x = 5, -4