Question

use the change of base formula to rewrite the logarithm in terms of the natural logarithm:
\\(\log_{9}(36) = \\)

use a calculator to evaluate the logarithm. round to four decimal places.

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Explanation:

Step1: Recall Change of Base Formula

The change of base formula for logarithms is $\log_b(a) = \frac{\ln(a)}{\ln(b)}$ (when converting to natural logarithm, where $\ln$ is the natural logarithm with base $e$). For $\log_9(36)$, we have $b = 9$ and $a = 36$. So, applying the formula, we get $\log_9(36)=\frac{\ln(36)}{\ln(9)}$.

Step2: Evaluate the Expression

First, calculate $\ln(36)$ and $\ln(9)$ using a calculator. $\ln(36)\approx3.5835$ and $\ln(9)\approx2.1972$. Then, divide these two values: $\frac{3.5835}{2.1972}\approx1.6309$.

Answer:

First, the rewritten form using natural logarithm is $\frac{\ln(36)}{\ln(9)}$. When evaluated and rounded to four decimal places, the value is $1.6309$.