Approximate the logarithm using the properties of logarithms, given $\log _{b} 2 \approx 0.3562, \log _{b} 3 \approx 0.5646$, and $\log _{b} 5 \approx 0.8271$. (Round your answer to four decimal places.)
\[
\log _{b} 75
\]
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Answer
Hence, the approximate value of $\log_b 75$ is $\boxed{2.2188}$.
Step 1 ：We are given the values of $\log_b 2$, $\log_b 3$, and $\log_b 5$ as 0.3562, 0.5646, and 0.8271 respectively.
Step 2 ：We can express 75 as $3 \times 5^2$.
Step 3 ：Using the properties of logarithms, we can express $\log_b 75$ as $\log_b 3 + 2 \times \log_b 5$.
Step 4 ：Substituting the given values, we get $\log_b 75 = 0.5646 + 2 \times 0.8271$.
Step 5 ：Solving the above expression, we get $\log_b 75 = 2.2188$.
Step 6 ：Hence, the approximate value of $\log_b 75$ is $\boxed{2.2188}$.
