Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

6. Exponential & Logarithmic Functions

Properties of Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Properties of Logarithms

Previous problemNext problem

2:29 minutes

Problem 59

Textbook Question

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

Identify the logarithm expression: \( \log_4 0.863 \).

Use the change of base formula: \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a common base, typically 10 or \( e \).

Choose the common logarithm (base 10) for calculation: \( \log_4 0.863 = \frac{\log_{10} 0.863}{\log_{10} 4} \).

Use a calculator to find \( \log_{10} 0.863 \) and \( \log_{10} 4 \).

Divide the result of \( \log_{10} 0.863 \) by \( \log_{10} 4 \) to get the final value.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above

Video duration:

Play a video:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form b^y = x, where b is the base, y is the exponent, and x is the result. The logarithm log_b(x) answers the question: 'To what power must the base b be raised to obtain x?' Understanding logarithms is essential for evaluating expressions like log4(0.863).

Change of Base Formula

The Change of Base Formula allows us to convert logarithms from one base to another, which is particularly useful when a calculator only computes logarithms in base 10 (common logarithm) or base e (natural logarithm). The formula is log_b(a) = log_k(a) / log_k(b), where k is any positive number. This enables the evaluation of log4(0.863) using a calculator by converting it to a more manageable base.

Common and Natural Logarithms

Common logarithms (log) use base 10, while natural logarithms (ln) use base e (approximately 2.718). These logarithms are widely used in various applications, including scientific calculations and financial modeling. Recognizing when to use each type is crucial for accurately evaluating logarithmic expressions and understanding their properties.