Solving Exponential Equations
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.
||e2x - 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.
|e2x - 6
e2x - 6
2x - 6
3x - 6
|= e -(x - 9)
= e - x + 9
= -x + 9
So, the solution of
is x = 5.
Recall the definition of a negative
The point (2, 25) lies on the graph of the exponential function y
Find the base, b, of this exponential function.
This results in the original ordered pair, (2, 25), as expected.
|Since (2, 25) lies on the graph, we know that x
= 2 when y = 25 in the exponential function.
|Replace y with 25 and x with 2.
Write 25 as 52.
Since the exponents are the same, the bases
So, the base, b, of the exponential function is 5.
We can now write the exponential function:
As a check, find f(2).
Note that if 52 = b2 then b could
also be -5.
However, we are working with an
That means the base,
b, must be such that b > 0 and b
Thus, b must be positive 5.