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

Introduction to Logarithms

College Algebra

6. Exponential & Logarithmic Functions

Introduction to Logarithms

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3:06 minutes

Problem 97

Textbook Question

Given that log￬10 2 ≈ 0.3010 and log￬10 3 ≈ 0.4771, find each logarithm without using a calculator. log￬10 9/4

Verified step by step guidance

Recognize that \( \log_{10} \frac{9}{4} \) can be rewritten using the quotient rule for logarithms: \( \log_{10} 9 - \log_{10} 4 \).

Express \( \log_{10} 9 \) as \( \log_{10} (3^2) \), which simplifies to \( 2 \cdot \log_{10} 3 \) using the power rule for logarithms.

Express \( \log_{10} 4 \) as \( \log_{10} (2^2) \), which simplifies to \( 2 \cdot \log_{10} 2 \) using the power rule for logarithms.

Substitute the given values: \( \log_{10} 3 \approx 0.4771 \) and \( \log_{10} 2 \approx 0.3010 \) into the expressions for \( \log_{10} 9 \) and \( \log_{10} 4 \).

Calculate \( 2 \cdot \log_{10} 3 - 2 \cdot \log_{10} 2 \) to find \( \log_{10} \frac{9}{4} \).

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Key Concepts

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

Properties of Logarithms

Logarithms have several key properties that simplify calculations. The quotient rule states that log_b(a/c) = log_b(a) - log_b(c), allowing us to break down complex logarithmic expressions into simpler parts. This property is essential for solving the given problem, as it enables the calculation of log₁₀(9/4) by separating it into log₁₀(9) and log₁₀(4).

Change of Base Formula

The change of base formula allows us to express logarithms in terms of logarithms of a different base. Specifically, log_b(a) can be rewritten as log_k(a) / log_k(b) for any positive k. This concept is useful when dealing with logarithms that are not easily computable, as it provides a method to relate them to known values, such as log₁₀(2) and log₁₀(3) in this case.

Logarithmic Values of Powers

Understanding how to compute logarithms of powers is crucial. For instance, log₁₀(9) can be expressed as log₁₀(3²), which simplifies to 2 * log₁₀(3) due to the power rule of logarithms. Similarly, log₁₀(4) can be expressed as log₁₀(2²). Recognizing these relationships allows for easier calculations when finding log₁₀(9/4).