To evaluate the expression \(\log_5 26\) using the change of base formula, we can apply the formula as follows:
\[ \log_b a = \frac{\log_k a}{\log_k b} \]
where \(k\) can be any base. In this case, we will use base \(e\) (natural logarithm, denoted as \(\ln\)) for consistency since we already have a value for \(\log_{10} 26\).
Given:
- \(\log_e 26 \approx 2.966\)
We need to find \(\log_5 26\):
\[ \log_5 26 = \frac{\log_e 26}{\log_e 5} \]
Now, you would need to find \(\log_e 5\). If not given, we can find it using common logarithms:
Using the change of base formula, we can find approximately:
\[ \log_e 5 = \frac{\log_{10} 5}{\log_{10} e} \]
Where typically:
- \(\log_{10} 5 \approx 0.699\)
- \(\log_{10} e \approx 0.434\)
So:
\[ \log_e 5 \approx \frac{0.699}{0.434} \approx 1.61 \]
Now substituting back into the equation for \(\log_5 26\):
\[ \log_5 26 = \frac{2.966}{1.61} \approx 1.84 \]
Therefore, \(\log_5 26\) is approximately 1.84.
If you need a more precise value, you'd calculate the exact value of \(\log_e 5\) using a scientific calculator or find it directly.