An exponential equation is an equation in which the unknown appears in the exponent.
To solve an exponential equation, we will take into account:
Properties of Powers
- a > 0
- If
then 
Solving Exponential Equations
Case 1: Both Sides Can Be Expressed in the Same Base
Perform the necessary operations so that both sides have the same base, so that we can equate the exponents.
Examples
1 
We rewrite the right side as 
and decompose the number 
Since 
, then:
We equate the exponents:
2 
We transform the roots into powers with fractional exponents and equate the exponents:
We solve the resulting equation:
3 
We factor out 
We apply the negative power law and perform the operations and solve for
We rewrite the equation with the same base and equate the exponents:
Case 2: The Sum of Terms of a Geometric Sequence
If we have the sum of 
terms of a geometric sequence, we apply the formula:
Example
Applying the formula for the sum of the terms of a geometric sequence:
We solve for and express both sides with the same base:
Case 3: Change of Variable
When we have a more complex equation, we can use a change of variable.
Examples
1 
First, we apply the property of the product of powers to remove the sum in the exponent.
We apply the power of a power property:
We perform the change of variable 
Factoring the equation and solving:
We undo the change of variable:
2 
We apply the properties of powers of products or quotients to remove additions or subtractions in the exponents:
We perform the change of variable 
We multiply both sides by 
We factor and solve the equation:
We undo the change of variable:
From the second equation, we do not obtain a solution.
3 
We decompose into factors 
and
We perform the change of variable:
We undo the change of variable only with the positive solution.
Since we cannot equate exponents, we take logarithms on both sides and on the first side we apply the property:
We solve for 
For the other solution with a negative sign, we would not have a solution because when we apply logarithms on the right side, we would find the logarithm of a negative number, which does not exist.
Case 4: Both Sides Cannot Be Expressed with the Same Base
To solve for an unknown that is in the exponent of a power, we take logarithms whose base is the base of the power.
Example
1 
We take logarithms on both sides:
We apply the logarithm of a power property:
Since 
We solve for
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