$$ 1 \leftarrow \quad $$ Use the change-of-base theorem to find the logarithm. $$ \log _{4} 9 $$

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To find the logarithm $$ \log_{4} 9 $$ using the change-of-base theorem, we can use the formula:

$$ \log_{a} b = \frac{\log_{c} b}{\log_{c} a} $$

where $$ a $$, $$ b $$, and $$ c $$ are positive real numbers and $$ c \neq 1 $$.

In this case, we want to find $$ \log_{4} 9 $$, so we can use the change-of-base theorem with base $$ c = 10 $$ (common logarithm) as follows:

$$ \log_{4} 9 = \frac{\log_{10} 9}{\log_{10} 4} $$

Now, we can calculate the values of $$ \log_{10} 9 $$ and $$ \log_{10} 4 $$ and then divide them to find $$ \log_{4} 9 $$.
The logarithm $$ \log_{4} 9 $$ is approximately equal to 1.584963.
Use the change-of-base theorem with base 10 to find $$ \log_{4} 9 $$, which is approximately 1.585.

Use the change-of-base theorem to find the logarithm.

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