Question

Use the change of base formula to find the value of the following logarithm. Do not round logarithms in the change of base formula.
log Subscript 8 Baseline 12
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Part 1
log Subscript 8 Baseline 12 equals
  
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​(Simplify your answer. Do not round until the final answer. Then round to four decimal places as​ needed.)

Answer 1

To find the value of \( \log_8 12 \) using the change of base formula, we can use the following relationship:

\[
\log_b a = \frac{\log_k a}{\log_k b}
\]

where \( k \) can be any positive number (commonly 10 or \( e \)). In this case, let's choose \( k = 10 \) (common logarithm) for simplicity:

\[
\log_8 12 = \frac{\log_{10} 12}{\log_{10} 8}
\]

Now, we'll calculate \( \log_{10} 12 \) and \( \log_{10} 8 \).

1. **Calculating \( \log_{10} 12 \)**:
Using the logarithm:

\[
\log_{10} 12 \approx 1.07918
\]

2. **Calculating \( \log_{10} 8 \)**:
Since \( 8 = 2^3 \):

\[
\log_{10} 8 = \log_{10} (2^3) = 3 \cdot \log_{10} 2
\]
Using \( \log_{10} 2 \approx 0.30103 \):

\[
\log_{10} 8 \approx 3 \cdot 0.30103 \approx 0.90309
\]

Now substituting back into the change of base formula:

\[
\log_8 12 = \frac{\log_{10} 12}{\log_{10} 8} \approx \frac{1.07918}{0.90309}
\]

Calculating this gives:

\[
\log_8 12 \approx 1.1956
\]

Finally, rounding to four decimal places:

\[
\log_8 12 \approx 1.1956
\]

Thus, rounded to four decimal places, \( \log_8 12 \) is:

\[
\boxed{1.1956}
\]