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 \( \log_{10} 8 \).
- Calculating \( \log_{10} 12 \): Using the logarithm:
\[ \log_{10} 12 \approx 1.07918 \]
- 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:
\[ \boxed{1.1956} \]