Question

evaluate the expression.
\log_{0.4} 14.1 \approx \square
(type an integer or a decimal. do not round until the fina

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 $c = 10$ or $c=e$). Here, $b = 0.4$ and $a=14.1$, so $\log_{0.4}14.1=\frac{\log 14.1}{\log 0.4}$ (using common logarithm with base 10).

Step2: Calculate numerator and denominator

First, calculate $\log 14.1\approx1.1492$ (using calculator for base 10 logarithm of 14.1). Then, calculate $\log 0.4\approx - 0.3979$ (using calculator for base 10 logarithm of 0.4).

Step3: Divide the two values

Now, divide the numerator by the denominator: $\frac{1.1492}{-0.3979}\approx - 2.888$.

Answer:

$-2.89$ (rounded to two decimal places, or more precise value as per calculation)