In Exercises 71–78, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log₁₄ 87.5

Verified step by step guidance

1. Identify the logarithm you need to evaluate: \( \log_{14} 87.5 \), which means the logarithm of 87.5 with base 14.
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): \( \log_{14} 87.5 = \frac{\log_{10} 87.5}{\log_{10} 14} \).
4. Use a calculator to find the values of \( \log_{10} 87.5 \) and \( \log_{10} 14 \) separately, making sure to keep at least four decimal places.
5. Divide the value of \( \log_{10} 87.5 \) by the value of \( \log_{10} 14 \) to get the final result for \( \log_{14} 87.5 \).

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 14 of 87.5 means finding the exponent x such that 14^x = 87.5. Understanding the relationship between exponents and logarithms is fundamental.

Change of Base Formula

Since calculators typically only compute logarithms with base 10 (common logs) or base e (natural logs), the change of base formula allows conversion: log_b(a) = log_c(a) / log_c(b), where c is 10 or e. This formula enables evaluation of logarithms with any base using a calculator.

Using a Calculator for Logarithms

Calculators can compute common logarithms (log base 10) and natural logarithms (log base e). To find log base 14 of 87.5, use the change of base formula with either log or ln functions on the calculator, then round the result to four decimal places as required.