AP Precalculus Practice Test: Unit 2 Question #32 Solving a Power Equals a Power Equation

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To solve a **power equals power equation**, where both sides of the equation are exponential expressions, you can follow these steps:

### Step 1: Set the Bases Equal (If Possible)
If both sides of the equation have the same base, you can set the exponents equal to each other. This is because if the bases are the same and the powers are equal, the exponents must be equal as well.

For example:
\[
a^x = a^y \quad \Rightarrow \quad x = y \text{ (if } a \text{ is positive and not equal to 1)}
\]

### Step 2: Rewrite the Equation if the Bases Are Different
If the bases on both sides are different, try to express them with the same base. This may involve factoring or using logarithms.

For example:
\[
2^x = 8 \quad \text{can be rewritten as} \quad 2^x = 2^3
\]
Then, by equating the exponents:
\[
x = 3
\]

### Step 3: Solve for the Exponent
Once the equation is rewritten with the same base, solve for the exponent by setting the exponents equal to each other.

### Example 1: Solve \( 3^{2x} = 3^5 \)

1. **Set the bases equal**: Since both sides of the equation have the same base (3), you can set the exponents equal to each other.
\[
2x = 5
\]

2. **Solve for \( x \)**:
\[
x = \frac{5}{2}
\]

So, the solution is \( x = \frac{5}{2} \).

### Example 2: Solve \( 5^{x+2} = 5^4 \)

1. **Set the bases equal**: The bases are both 5, so we can set the exponents equal to each other.
\[
x + 2 = 4
\]

2. **Solve for \( x \)**:
\[
x = 4 - 2 = 2
\]

So, the solution is \( x = 2 \).

### Example 3: Solve \( 2^{x+1} = 8 \)

1. **Rewrite 8 as a power of 2**:
\[
8 = 2^3
\]
So, the equation becomes:
\[
2^{x+1} = 2^3
\]

2. **Set the exponents equal**:
\[
x + 1 = 3
\]

3. **Solve for \( x \)**:
\[
x = 3 - 1 = 2
\]

So, the solution is \( x = 2 \).

### Example 4: Solve \( 4^{x} = 16^{x-1} \)

1. **Rewrite both sides with the same base**:
\[
4 = 2^2, \quad 16 = 2^4
\]
The equation becomes:
\[
(2^2)^x = (2^4)^{x-1}
\]

2. **Simplify both sides**:
\[
2^{2x} = 2^{4(x-1)}
\]

3. **Set the exponents equal**:
\[
2x = 4(x - 1)
\]

4. **Solve for \( x \)**:
\[
2x = 4x - 4 \quad \Rightarrow \quad 4 = 2x \quad \Rightarrow \quad x = 2
\]

So, the solution is \( x = 2 \).

### General Steps for Solving Power Equals Power Equations:
1. **Check if the bases are the same**. If they are, set the exponents equal.
2. **If the bases are different**, try rewriting the equation with the same base.
3. **Solve for the exponent** once the bases are the same.
4. **Check for extraneous solutions** (in some cases, such as with negative bases).

These steps will help you solve power equals power equations successfully.

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Nick Perich
Norristown Area High School
Norristown Area School District
Norristown, Pa

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