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