Question

evaluate the expression
\log_{0.4} 14.1 \approx \square
(type an integer or a decimal do not round until the final answer then round to four decimal places as needed )

Explanation:

Step1: Use Change of Base Formula

The change of base formula for logarithms is $\log_b a = \frac{\log_c a}{\log_c b}$ (we can use natural logarithm or common logarithm). Let's use natural logarithm (ln) here. So, $\log_{0.4} 14.1 = \frac{\ln 14.1}{\ln 0.4}$.

Step2: Calculate the Numerator and Denominator

First, calculate $\ln 14.1$. Using a calculator, $\ln 14.1 \approx 2.646174$. Then, calculate $\ln 0.4$. Using a calculator, $\ln 0.4 \approx -0.916291$.

Step3: Divide the Two Results

Now, divide the numerator by the denominator: $\frac{2.646174}{-0.916291} \approx -2.8879$.

Answer:

-2.8879