Solving Exponential Equations
An exponential equation is an equation where one or more of the
exponents contains a variable. Some types of exponential equations can
be solved using the following property.
Property â€” Exponential Equality
If b^{x} = b^{y}, then x = y. Here, b > 0 and b
≠ 1.
Note:
Recall that for a onetoone
function, two different inputs
always result in two different
outputs.
That is, in a onetoone function,
each output value corresponds to
exactly one input value.
This property is a direct consequence of the fact that exponential functions
are onetoone functions. That is, if two exponential functions have the
same output, bx and by, then their inputs, x and y, must be equal.
Example 1
Solve: 25^{ 5x8} = 625
Solution
First, write each expression using the same base, 5.
On the left, use the Power of a Power Property.
Use the Exponential Equality Property to set
the exponents equal to each other.

25^{ 5x  8} (5^{2})^{ 5x  8}
5^{ 10x  16}
10x  16 
= 625 = 5^{4}
= 5^{4}
= 4 
Add 16 to both sides.
Divide both sides by 10. 
10x x 
= 20 = 2 
We can check the solution by replacing x with 2 in the original equation
and simplifying.
Original equation: 
Is
Is
Is 
25^{ 5x  8}
25^{ 5 Â· 2  8}
25^{2}
625 
= 625 = 625 ?
= 625 ?
= 625 ? Yes 
So, the solution of 25^{ 5x  8 }= 625 is x = 2.
Note:
To write each base in exponential form,
first find its prime factorization.
For example:
25 = 5 Â· 5 = 5^{2}
625 = 5 Â· 5
Â· 5 Â· 5 = 5^{4}
The Power of a Power Property says
(xm)n 5 xmn.
Example 2
Solve: 16^{3x + 1} = 32^{4x}
Solution
Write each expression with the same base, 2.
Use the Power of a Power Property.
Use the Exponential Equality Property to set
the exponents equal to each other.

16^{3x + 1} (2^{4})^{3x + 1}
2^{12x + 4}
12x + 4 
= 32^{4x} = (2^{5})^{4x}
= 2^{20x}
= 20x 
Subtract 12x from both sides. 
4 
= 8x 
Divide both sides by 8.


= x 
We can check the solution in the usual way.
So, the solution of 16^{3x + 1} = 32^{4x} is
Note:
This method can only be used if we
write each side of the equation as an
exponential expression using the same
base. If not, we must use a different
method.
For example, we can use this method
solve 2^{x} = 8 but not to
solve 2^{x} = 9.
Recall that
Thus,
