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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1:33 minutes

Problem 43

Textbook Question

Solve each problem. Use a calculator to find an approximation for each logarithm. log 3.984

Verified step by step guidance

Identify the base of the logarithm. In this case, it is assumed to be base 10, as no base is specified.

Use the change of base formula if necessary: \( \log_b a = \frac{\log_c a}{\log_c b} \). Here, \( b = 10 \) and \( a = 3.984 \).

Enter the value 3.984 into a scientific calculator and use the 'log' function to find \( \log_{10} 3.984 \).

The calculator will provide an approximate decimal value for \( \log_{10} 3.984 \).

Interpret the result as the power to which the base 10 must be raised to obtain the number 3.984.

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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 inverse operation to exponentiation, meaning it answers the question: to what exponent must a base be raised to produce a given number? For example, log_b(a) = c implies that b^c = a. Understanding logarithms is essential for solving problems involving exponential growth or decay, as well as for simplifying complex calculations.

Common Logarithm

The common logarithm is a logarithm with base 10, often denoted as log(x) or log_10(x). It is widely used in scientific calculations and is particularly useful for simplifying multiplication and division into addition and subtraction. Knowing how to use a calculator to compute common logarithms is crucial for solving logarithmic problems efficiently.

Calculator Functions

Most scientific calculators have built-in functions for calculating logarithms, typically labeled as 'log' for common logarithms and 'ln' for natural logarithms (base e). Familiarity with these functions allows students to quickly find logarithmic values without manual computation, which is essential for solving problems that require precise numerical approximations.