Question

lesson 25 - logarithmic functions and equations
score: 6/100 answered: 2/19
question 3
evaluate the following logarithms using a calculator and the change of base formula.
\\(\log_{4}(0.5) = \square\\)
\\(\log_{5}(2.94) = \square\\)
\\(\log_{3}(4.71) = \square\\)
round your answer to two decimals.
question help: message instructor

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 can use base 10 or natural logarithm).

Step2: Evaluate $\log_4(0.5)$

Using the formula, $\log_4(0.5) = \frac{\log(0.5)}{\log(4)}$ (using base 10 logarithm).
$\log(0.5) \approx -0.3010$, $\log(4) \approx 0.6021$.
So, $\frac{-0.3010}{0.6021} \approx -0.50$.

Step3: Evaluate $\log_5(2.94)$

Using the formula, $\log_5(2.94) = \frac{\log(2.94)}{\log(5)}$.
$\log(2.94) \approx 0.4683$, $\log(5) \approx 0.6990$.
So, $\frac{0.4683}{0.6990} \approx 0.67$.

Step4: Evaluate $\log_3(4.71)$

Using the formula, $\log_3(4.71) = \frac{\log(4.71)}{\log(3)}$.
$\log(4.71) \approx 0.6729$, $\log(3) \approx 0.4771$.
So, $\frac{0.6729}{0.4771} \approx 1.41$.

Answer:

$\log_4(0.5) \approx -0.50$, $\log_5(2.94) \approx 0.67$, $\log_3(4.71) \approx 1.41$