Problem

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.) \[ \log _{b} 75 \] Need Help? Read It Submit Answer

Solution

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 $\boxed{2.2188}$.

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