Solving an exponential equation
An exponential equation is an equation in which the unknown appears as an exponent.
Generally speaking, we solve the less complex exponential equations through applying the laws of exponentiation.
For the more complicated exponential equations, we refer to the properties and the use of logarithms (refer to lesson 17).
In the exponential equation
y = ax where a > 0 and a 
1,
a is called the base of the exponent and the power
x is referred to as the logarithm [of y] in base
a.
Exponentiation laws
Law 1 am X an = am + n
Example
a3 X a2 = a3+2 = a5
Law 2 (am)n = amXn
Example
(a3)2 = a3X2 = a6
Law 3 
= am-n if m > n and a
0
Example
= a5-3 = a2
Law 4
= 
if m < n and a
0
Example
= 
= 
Law 5 (a X b)m = am X bm
Example
(a X b)3 = a3 X b3
Law 6 (
)m
= 
where b 0
Example
()2 = 
Example 1
Solve the following equation; 52x = 25
Solution
It is simply a question of re-writing the terms of the equation in an exponential form with the same base.
Following this guideline, 52x = 52 and so 2x = 2
Thus x = 1
Example 2
Solve; 8x-1 = 
Solution
(23)x-1 = 1/22
(23)x-1 = 2-2
23(x-1) = 2-2
3(x-1) = -2
x = 1/3
Before proceeding to the questions, ensure that you have a pencil and some paper handy.
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