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

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2:20 minutes

Problem 33

Textbook Question

Use a calculator to find an approximation to four decimal places for each logarithm. log₂/₃ 5/8

Verified step by step guidance

Step 1: Understand that the problem requires finding the logarithm of \( \frac{5}{8} \) with base \( \frac{2}{3} \).

Step 2: Use the change of base formula for logarithms: \( \log_b a = \frac{\log_c a}{\log_c b} \), where \( c \) is a base that your calculator can handle, typically 10 or \( e \).

Step 3: Apply the change of base formula: \( \log_{\frac{2}{3}} \frac{5}{8} = \frac{\log \frac{5}{8}}{\log \frac{2}{3}} \).

Step 4: Use a calculator to find \( \log \frac{5}{8} \) and \( \log \frac{2}{3} \) using the base 10 logarithm function.

Step 5: Divide the result of \( \log \frac{5}{8} \) by \( \log \frac{2}{3} \) to find the approximate value of the logarithm to four decimal places.

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Key Concepts

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

Logarithms

A logarithm is the power to which a base must be raised to produce a given number. In the expression log_b(a), b is the base, and a is the number. Understanding logarithms is essential for solving equations involving exponential growth or decay, as well as for converting multiplicative relationships into additive ones.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, which is particularly useful when using calculators that typically only compute logarithms in base 10 or base e. The formula is log_b(a) = log_k(a) / log_k(b), where k is any positive number. This enables the calculation of logarithms with bases that are not directly supported by the calculator.

Approximation and Rounding

Approximation involves estimating a value that is close to the exact answer, often used in calculations where precision is not critical. Rounding is the process of reducing the number of significant digits in a number. In this context, finding an approximation to four decimal places means calculating the logarithm and then rounding the result to four digits after the decimal point for clarity and simplicity.