Question

use common logarithms or natural logarithms and a calculator to evaluate the expression.

\\(\log_{0.9} 25.8\\)

evaluate the expression.

\\(\log_{0.9} 25.8 \approx \square\\)

(type an integer or a decimal. do not round until the final answer. then round to four decimal places as needed.)

Explanation:

State the given expression and goal

We need to evaluate the logarithmic expression:

$$ \log_{0.9} 25.8 $$

Our goal is to approximate this value to four decimal places using the change-of-base formula.

Apply the change-of-base formula

The change-of-base formula allows us to rewrite any logarithm using common logarithms (base 10) or natural logarithms (base \(e\)):

$$ \log_b a = \frac{\ln a}{\ln b} = \frac{\log a}{\log b} $$

Using natural logarithms, we rewrite the expression as:

$$ \log_{0.9} 25.8 = \frac{\ln 25.8}{\ln 0.9} $$

Calculate the numerator and denominator

We evaluate the natural logarithms of the numerator and denominator:

$$ \ln 25.8 \approx 3.250374 $$

$$ \ln 0.9 \approx -0.105361 $$

Divide the values

Now, we divide the two calculated values:

$$ \frac{3.250374}{-0.105361} \approx -30.849998 $$

Round to four decimal places

Rounding the result to four decimal places:

$$ -30.8500 $$

Answer:

Evaluate the expression.
\(\log_{0.9}25.8 \approx\) <blank>-30.8500</blank>
(Type an integer or a decimal. Do not round until the final answer. Then round to four decimal places as needed.)