Use the change of base formula to compute $$ \log _{4} 6 $$. Round your answer to the nearest thousandth.

Question

UpStudy AI · Accepted Answer

To compute $$ \log_{4} 6 $$ using the change of base formula, we can use the 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:

$$
\log_{4} 6 = \frac{\log_{10} 6}{\log_{10} 4}
$$

Now, we need to calculate $$ \log_{10} 6 $$ and $$ \log_{10} 4 $$.

Using a calculator:

$$
\log_{10} 6 \approx 0.778151
$$
$$
\log_{10} 4 \approx 0.602060
$$

Now we can substitute these values into the formula:

$$
\log_{4} 6 = \frac{0.778151}{0.602060}
$$

Calculating this gives:

$$
\log_{4} 6 \approx 1.292481
$$

Rounding to the nearest thousandth, we find:

$$
\log_{4} 6 \approx 1.292
$$

Thus, the final answer is:

$$
\boxed{1.292}
$$
The value of $$ \log_{4} 6 $$ rounded to the nearest thousandth is 1.292.

Use the change of base formula to compute \( \log _{4} 6 \). Round your answer to the nearest thousandth.