Question

evaluate the following logarithms using a calculator and the change of base formula.
\\(\log_{2}(0.382) = \square\\)
\\(\log_{6}(1.38) = \square\\)
\\(\log_{5}(92.1) = \square\\)
round your answer to two decimals.
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Explanation:

Step1: Recall Change of Base Formula

The Change of Base Formula is $\log_b(a) = \frac{\log_c(a)}{\log_c(b)}$, where $c$ can be 10 or $e$ (for calculator use). We'll use base 10.

Step2: Evaluate $\log_2(0.382)$

Using the formula: $\log_2(0.382) = \frac{\log(0.382)}{\log(2)}$. Calculate $\log(0.382) \approx -0.419$, $\log(2) \approx 0.3010$. Then $\frac{-0.419}{0.3010} \approx -1.39$.

Step3: Evaluate $\log_6(1.38)$

Using the formula: $\log_6(1.38) = \frac{\log(1.38)}{\log(6)}$. $\log(1.38) \approx 0.140$, $\log(6) \approx 0.7782$. Then $\frac{0.140}{0.7782} \approx 0.18$.

Step4: Evaluate $\log_5(92.1)$

Using the formula: $\log_5(92.1) = \frac{\log(92.1)}{\log(5)}$. $\log(92.1) \approx 1.9643$, $\log(5) \approx 0.6990$. Then $\frac{1.9643}{0.6990} \approx 2.81$.

Answer:

$\log_2(0.382) \approx -1.39$, $\log_6(1.38) \approx 0.18$, $\log_5(92.1) \approx 2.81$