Lesson Objectives
• Demonstrate an understanding of exponential functions
• Learn how to solve an exponential equation with like bases

## How to Solve Exponential Equations with Like Bases

An exponential equation is an equation with a variable in the exponent.
f(x) = ax
for a > 0, a ≠ 1
To solve an exponential equation with like bases on each side, we use the following rule:
ax = ay
In simpler cases, we can use the above rule to solve exponential equations. In other cases, we will need to rely on logarithms, which we will learn about over the course of the next few lessons. To solve an exponential equation with like bases, we can use the following steps.
• Make sure each side has the same base
• In some cases, we can rewrite our expression using the rules of exponents
• Simplify the exponents
• Set the exponents equal
• Solve the resulting equation
• Check
Let's look at a few examples.
Example 1: Solve each equation. $$3^{2x + 1}=27$$ Step 1) Make sure each side has the same base.
In this case, the left side has a base of 3, while the right side has a base of 27. Using the rules of exponents, we can rewrite 27 as 33. $$3^{2x + 1}=3^3$$ Step 2) Simplify the exponents.
In this case, the exponents are simplified.
Step 3) Set the exponents equal. $$2x + 1=3$$ Step 4) Solve the resulting equation. $$2x + 1=3$$ Subtract 1 away from each side: $$2x=2$$ Divide each side by 2: $$x=1$$ Step 5) Check $$3^{2x + 1}=3^3$$ $$3^{2(1) + 1}=3^3$$ $$3^{3}=3^3$$ $$\require{color}27=27 \hspace{.2em}\color{green}{✔}$$ Example 2: Solve each equation. $$25^{-3x-3}\cdot 5^{-2x - 2}=625$$ Step 1) Make sure each side has the same base.
In this case, we will write each base as 5, using the rules of exponents. $$5^{2(-3x-3)}\cdot 5^{-2x - 2}=5^4$$ $$5^{-6x - 6}\cdot 5^{-2x - 2}=5^4$$ Step 2) Simplify the exponents.
On the left side of the equation, we can use our product rule for exponents. $$5^{-6x - 6 - 2x - 2}=5^4$$ $$5^{-8x - 8}=5^4$$ Step 3) Set the exponents equal. $$-8x - 8=4$$ Step 4) Solve the resulting equation. $$-8x - 8=4$$ Add 8 to each side: $$-8x=12$$ Divide each side by -8: $$x=-\frac{12}{8}=-\frac{3}{2}$$ Step 5) Check $$25^{-3\cdot -\frac{3}{2}- 3}\cdot 5^{-2 \cdot -\frac{3}{2}- 2}=625$$ $$25^{\frac{9}{2}- 3}\cdot 5^{3 - 2}=625$$ $$25^{\frac{9}{2}- \frac{6}{2}}\cdot 5^1=625$$ $$25^{\frac{3}{2}}\cdot 5^{1}=625$$ $$125 \cdot 5=625$$ $$625=625 \hspace{.2em}\color{green}{✔}$$ Example 3: Solve each equation. $$216 \cdot 216^{3x}=36^{-3x - 1}$$ Step 1) Make sure each side has the same base. In this case, we will write each base as 6, using the rules of exponents. $$6^3 \cdot (6^{3})^{3x}=(6^2)^{-3x - 1}$$ Step 2) Simplify the exponents.
First, let's use our power-to-power rule. $$6^3 \cdot 6^{9x}=6^{-6x - 2}$$ Now, let's use our product rule for exponents on the left side of the equation. $$6^{9x + 3}=6^{-6x - 2}$$ Step 3) Set the exponents equal. $$9x + 3=-6x - 2$$ Step 4) Solve the resulting equation. $$9x + 3=-6x - 2$$ Add 6x to both sides: $$15x + 3=-2$$ Subtract 3 away from each side: $$15x=- 5$$ Divide both sides by 15: $$x=-\frac{5}{15}=-\frac{1}{3}$$ Step 5) Check $$216 \cdot 216^{3x}=36^{-3x - 1}$$ $$216 \cdot 216^{3\left(-\frac{1}{3}\right)}=36^{-3\left(-\frac{1}{3}\right) - 1}$$ $$216 \cdot 216^{-1}=36^{0}$$ $$1=1 \hspace{.2em}\color{green}{✔}$$

#### Skills Check:

Example #1

Solve each equation. $$36^{3x}=\frac{1}{6}$$

A
No Solution
B
All Real Numbers
C
$$x=-\frac{1}{6}$$
D
$$x=\frac{1}{3}$$
E
$$x=-6$$

Example #2

Solve each equation. $$16^{2x}=64^{-3x - 3}$$

A
No Solution
B
$$x=-5$$
C
$$x=-2$$
D
$$x=-\frac{9}{13}$$
E
$$x=\frac{3}{4}$$

Example #3

Solve each equation. $$9^{3x}=81^{-x}$$

A
All Real Numbers
B
No Solution
C
$$x=3$$
D
$$x=0$$
E
$$x=-\frac{1}{5}$$