In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log₅ 13

Verified step by step guidance

1. Recognize that the logarithm given is \( \log_5 13 \), which means the logarithm of 13 with base 5.
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): \( \log_5 13 = \frac{\log_{10} 13}{\log_{10} 5} \).
4. Use a calculator to find the values of \( \log_{10} 13 \) and \( \log_{10} 5 \) separately, making sure to keep enough decimal places for accuracy.
5. Divide the value of \( \log_{10} 13 \) by \( \log_{10} 5 \) to get \( \log_5 13 \), 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 5 of 13 means finding the exponent x such that 5^x = 13. Understanding the relationship between exponents and logarithms is fundamental.

Change of Base Formula

The change of base formula allows you to evaluate logarithms with any base using common (base 10) or natural (base e) logarithms: log_b(a) = log(a) / log(b). This is essential when calculators only provide log or ln functions, enabling calculation of log base 5 of 13.

Using a Calculator for Logarithms

Calculators typically have buttons for common logarithms (log base 10) and natural logarithms (ln). To find log base 5 of 13, use the change of base formula with either log or ln, then compute the values and divide, rounding the result to four decimal places as required.