Questions: Solve the exponential equation by expressing each side as a power of the same base and then equating the exponents. 3^(2x-16) = 9 The solution set is .

Solve the exponential equation by expressing each side as a power of the same base and then equating the exponents.
3^(2x-16) = 9

The solution set is .

Transcript text: Solve the exponential equation by expressing each side as a power of the same base and then equating the exponents. \[ 3^{2 x-16}=9 \] The solution set is $\square$ \}.

Solution

Solution Steps

Step 1: Express Each Side as a Power of the Same Base

The equation given is:

\[ 3^{2x - 16} = 9 \]

We know that $9$ can be expressed as a power of $3$. Specifically, $9 = 3^2$. Therefore, we can rewrite the equation as:

\[ 3^{2x - 16} = 3^2 \]

Step 2: Equate the Exponents

Since the bases are the same, we can equate the exponents:

\[ 2x - 16 = 2 \]

Step 3: Solve for $x$

To solve for $x$, add $16$ to both sides of the equation:

\[ 2x - 16 + 16 = 2 + 16 \]

\[ 2x = 18 \]

Now, divide both sides by $2$:

\[ x = \frac{18}{2} \]

\[ x = 9 \]

Final Answer

The solution set is $\boxed{x = 9}$.

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