Question
Use the change-of-base formula to approximate . (Round to 3 decimal places.)

Question

Question Use the change-of-base formula to approximate $$ \log _{3} 24 $$. (Round to 3 decimal places.)

UpStudy AI · Accepted Answer

To approximate $$ \log_{3} 24 $$ using the change-of-base formula, we can use the following formula:

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

where $$ k $$ can be any positive number (commonly 10 or $$ e $$). Here, we will use base 10 for convenience.

Applying the change-of-base formula, we have:

$$
\log_{3} 24 = \frac{\log_{10} 24}{\log_{10} 3}
$$

Now, we need to calculate $$ \log_{10} 24 $$ and $$ \log_{10} 3 $$.

Using a calculator:

$$
\log_{10} 24 \approx 1.3802
$$
$$
\log_{10} 3 \approx 0.4771
$$

Now, substituting these values into the formula:

$$
\log_{3} 24 \approx \frac{1.3802}{0.4771} \approx 2.895
$$

Rounding to three decimal places, we find:

$$
\log_{3} 24 \approx 2.895
$$

Thus, the approximate value of $$ \log_{3} 24 $$ is $$ \boxed{2.895} $$.
The approximate value of $$ \log_{3} 24 $$ is 2.895.

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