Question

use the change of base formula to rewrite the logarithm in terms of the natural logarithm:
\\(\log_{6}(77) = \square\\)
use a calculator to evaluate the logarithm. round to four decimal places.
\\(\square\\)

Explanation:

Step1: Recall Change of Base Formula

The Change of Base Formula for a logarithm $\log_b(a)$ is $\frac{\ln(a)}{\ln(b)}$ (when using natural logarithm) or $\frac{\log(a)}{\log(b)}$ (when using common logarithm). Here we need to rewrite $\log_6(77)$ in terms of natural logarithm, so we use $\log_b(a)=\frac{\ln(a)}{\ln(b)}$. Substituting $a = 77$ and $b = 6$, we get $\log_6(77)=\frac{\ln(77)}{\ln(6)}$.

Step2: Evaluate the Expression

First, calculate $\ln(77)$ and $\ln(6)$ using a calculator. $\ln(77)\approx4.343805$, $\ln(6)\approx1.791759$. Then divide them: $\frac{4.343805}{1.791759}\approx2.4247$.

Answer:

First part (rewritten form): $\frac{\ln(77)}{\ln(6)}$
Second part (evaluated value): $2.4247$