Solving Exponential Equations
Example 1
Solve:
Solution
Each side of the equation contains an expression with base e. However, in
the expression on the right, the base e is in the denominator.
Therefore, we use the definition of a negative exponent to write the
expression on the right with in the numerator.
Original equation.

e^{2x  6} 

Rewrite the right side.
Remove the parentheses.
Use the Exponential Equality Property.
Add x to both sides.
Add 6 to both sides.
Divide both sides by 3. 
e^{2x  6}
e^{2x  6}
2x  6
3x  6
3x
x 
= e^{ (x  9)}
= e^{  x + 9}
= x + 9
= 9
= 15
= 5 
So, the solution of
is x = 5.
Note:
Recall the definition of a negative
exponent:
Example 2
The point (2, 25) lies on the graph of the exponential function y
= b^{x}.
Find the base, b, of this exponential function.
Solution
Since (2, 25) lies on the graph, we know that x
= 2 when y = 25 in the exponential function. 
y 
= b^{x} 
Replace y with 25 and x with 2.
Write 25 as 5^{2}.
Since the exponents are the same, the bases
are equal.
So, the base, b, of the exponential function is 5.
We can now write the exponential function:
As a check, find f(2). 
25
5^{2}
5
f(x)
f(2) 
= b^{2}
= b^{2}
= b
= 5^{x}
= 5^{2}
= 25 
This results in the original ordered pair, (2, 25), as expected.
Note:
Note that if 5^{2} = b^{2} then b could
also be 5.
However, we are working with an
exponential function.
That means the base,
b, must be such that b > 0 and b
≠ 1.
Thus, b must be positive 5.
