Question

use the change of base formula to rewrite the logarithm in terms of the natural logarithm: \\(\log_{8}(23) = \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 logarithms is $\log_b(a) = \frac{\ln(a)}{\ln(b)}$, where $\ln$ is the natural logarithm. For $\log_8(23)$, we can apply this formula with $b = 8$ and $a = 23$. So, $\log_8(23)=\frac{\ln(23)}{\ln(8)}$.

Step2: Evaluate the Expression

Using a calculator, we first find the values of $\ln(23)$ and $\ln(8)$. $\ln(23)\approx3.1354942159$ and $\ln(8)\approx2.0794415417$. Then we divide these two values: $\frac{3.1354942159}{2.0794415417}\approx1.5076$.

Answer:

First part (rewritten using natural logs): $\frac{\ln(23)}{\ln(8)}$
Second part (evaluated, rounded to four decimals): $1.5076$