In Exercises 58–59, use common logarithms or natural logarithms and a calculator to evaluate to four decimal places. log₄ 0.863

Verified step by step guidance

1. Recognize that the problem asks for the logarithm of 0.863 with base 4, written as \(\log_{4} 0.863\).
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): \(\log_{4} 0.863 = \frac{\log_{10} 0.863}{\log_{10} 4}\).
4. Use a calculator to find the values of \(\log_{10} 0.863\) and \(\log_{10} 4\) separately, making sure to keep the values to at least four decimal places.
5. Divide the two logarithm values obtained in the previous step to get the value of \(\log_{4} 0.863\), rounded 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 4 of 0.863 means finding the exponent x such that 4^x = 0.863. 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 evaluating log base 4 of 0.863 using a calculator.

Using a Calculator for Logarithms

Calculators can compute common logarithms (log base 10) and natural logarithms (log base e). By applying the change of base formula, you input log(0.863) divided by log(4) or ln(0.863) divided by ln(4) to find the value. Rounding the result to four decimal places completes the evaluation.