SOLVING EXPONENTIAL EQUATIONS
Solve 5x  25
5x  25
log 5 (5x )  log 5 (25)
x  log 5 (5)=
log10 (25)
log10 (5)
Take log base 5 of each side so we
can move the exponent x in front of
the log.
Note: log5(5) = 1
x 1  2
x2
Solution set = {2}
Solve 5x = 1/25
5x = 0.04
log5(5x) = log5(0.04)
Take base-5 log of both sides.
xlog5(5) = log10(0.04)/log10(5)
Use Power Rule and change-of-base formula
x = log10(0.04)/log10(5)
Note: log5(5) = 1
x = -2
Solution set is {-2}
1
Solve 10x = 0.01
log10(10x) = log10(0.01)
[x]log10(10) = log10(0.01)/log10(10)
x = log10(0.01)/log10(10)
Take base-10 log on both sides.
Use Power Rule and change-of-base formula
Note: log10(10) = 1
x = -2
Solution set is {-2}
Solve 6 x  6
Solution
6x  6
log 6 (6 x )  log 6
x  log 6 (6) 
 6
log10 ( 6)
log10 (6)
x  1  0.5
x  0.5
Solution set is {0.5}
2
Solve 7 x  3 7
7x  3 7
 7
log  7 
x  log  7  
log 7  7 x   log 7
3
3
10
7
log10 (7)
x  1  1/ 3
x  1/ 3
Solution set is {1/3}
Solve 8x  1
8x  1
log 8  8x   log 8 1
x  log 8  8  
log10 (1)
log10 (8)
x 1  0
x0
Solution set is {0}.
3
Solve 16 x  8
Solution :
16 x  8
log16 (16 x )  log16 (8)
x  log16 (16) 
log10 (8)
log10 (16)
x  1  0.75
x  0.75
Solution set is {0.75}
4
5