In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log_0.1 17

Verified step by step guidance

1. Recognize that the logarithm given is \( \log_{0.1} 17 \), which means the logarithm of 17 with base 0.1.
2. Recall the change of base formula for logarithms: \( \log_a b = \frac{\log_c b}{\log_c a} \), where \( c \) can be any positive number (commonly 10 or \( e \)).
3. Apply the change of base formula using common logarithms (base 10): \( \log_{0.1} 17 = \frac{\log_{10} 17}{\log_{10} 0.1} \).
4. Use a calculator to find \( \log_{10} 17 \) and \( \log_{10} 0.1 \) separately, keeping the values to at least 5 decimal places for accuracy.
5. Divide the value of \( \log_{10} 17 \) by \( \log_{10} 0.1 \) to get the final result, then round your answer to four decimal places.

Key Concepts

Logarithms and Their Bases

A logarithm answers the question: to what power must the base be raised to produce a given number? In this problem, log base 0.1 of 17 means finding the exponent x such that 0.1^x = 17. Understanding how the base affects the logarithm is crucial for solving the problem.

Change of Base Formula

The change of base formula allows you to compute logarithms with any base using common (base 10) or natural (base e) logarithms: log_b(a) = log_c(a) / log_c(b). This is essential here because calculators typically only compute log base 10 or e, not base 0.1.

Using a Calculator for Logarithms

Calculators can evaluate common logarithms (log base 10) and natural logarithms (log base e) directly. By applying the change of base formula, you can use these functions to find logarithms with other bases and round the result to four decimal places as required.