Question

Find the exact value of the logarithm without using a calculator.
log Subscript 6 Baseline 36
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Part 1
log Subscript 6 Baseline 36equals
  
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​(Type an integer or a simplified​ fraction.)

Answer 1

To find the exact value of \( \log_6 36 \), we can use the change of base formula and the properties of logarithms.

First, we can express \( 36 \) in terms of base \( 6 \):

\[
36 = 6^2
\]

Now we can rewrite the logarithm:

\[
\log_6 36 = \log_6 (6^2)
\]

Using the logarithmic property \( \log_b (a^n) = n \cdot \log_b a \), we can simplify:

\[
\log_6 (6^2) = 2 \cdot \log_6 6
\]

Since \( \log_6 6 = 1 \):

\[
2 \cdot \log_6 6 = 2 \cdot 1 = 2
\]

Thus, the exact value of \( \log_6 36 \) is:

\[
\boxed{2}
\]