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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2:09 minutes

Problem 83

Textbook Question

Use the change-of-base theorem to find an approximation to four decimal places for each logarithm. See Example 8. . log_1/2 3

Verified step by step guidance

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

Apply the change-of-base formula to \( \log_{1/2} 3 \): \( \log_{1/2} 3 = \frac{\log_{10} 3}{\log_{10} (1/2)} \).

Use a calculator to find \( \log_{10} 3 \) and \( \log_{10} (1/2) \).

Divide the result of \( \log_{10} 3 \) by the result of \( \log_{10} (1/2) \).

Round the final result 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.

Change of Base Theorem

The Change of Base Theorem allows us to compute logarithms with bases other than 10 or e by converting them into a more manageable form. Specifically, it states that for any positive numbers a, b, and c (where a ≠ 1 and b ≠ 1), the logarithm can be expressed as log_b(a) = log_c(a) / log_c(b). This theorem is particularly useful when using calculators that only support certain bases.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are defined as log_b(a) = c if and only if b^c = a. They are essential in solving equations involving exponentials and are widely used in various fields, including science and engineering. Understanding the properties of logarithms, such as the product, quotient, and power rules, is crucial for manipulating and simplifying logarithmic expressions.

Approximation and Decimal Places

When calculating logarithms, especially with the change of base theorem, approximations are often necessary. This involves rounding the result to a specified number of decimal places, which is important for precision in applications. In this context, finding an approximation to four decimal places means ensuring that the result is accurate to the fourth digit after the decimal point, which can affect the outcome in practical scenarios.