Problem 25 - Precalculus with Limits

Question

Using Properties of Logarithms In Exercises \(21-36,\) find the exact value of the logarithmic expression without using a calculator. (If this is not possible, then state the reason.) $$\log _{4} 16^{2}$$

Step-by-Step Solution

Answer

The exact value of \( \log_{4} 16^{2} \) is 4.

Step 1: Apply the power rule property of logarithms

According to the power property of logarithms, we need to bring the power down as a multiplication. So the given expression \( \log_{4} 16^{2} \) can be written as \( 2 \log_{4} 16 \).

Step 2: Evaluate \( \log_{4} 16 \)

Since \( 4^{2} = 16 \), the value of \( \log_{4} 16 \) is 2.

Step 3: Substitute into expression

Substitute the value from step 2 into the expression from step 1, resulting in \( 2*2 \).

Step 4: Perform final calculation

The result of \( 2*2 \) is 4. This is the final answer.