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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1:51 minutes

Problem 29

Textbook Question

In Exercises 21–42, evaluate each expression without using a calculator. log7 √7

Verified step by step guidance

Recognize that the expression involves a logarithm with base 7: \( \log_7 \sqrt{7} \).

Recall the property of logarithms: \( \log_b (a^c) = c \cdot \log_b a \).

Express \( \sqrt{7} \) as an exponent: \( \sqrt{7} = 7^{1/2} \).

Apply the logarithm property: \( \log_7 (7^{1/2}) = \frac{1}{2} \cdot \log_7 7 \).

Use the fact that \( \log_7 7 = 1 \) because any logarithm of a number to its own base is 1.

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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 obtain a given number. In the expression log_b(a), b is the base, and a is the number. Understanding logarithms is essential for solving problems involving exponential relationships and for simplifying expressions involving powers.

Properties of Logarithms

Logarithms have several key properties that simplify calculations. For instance, the product property states that log_b(xy) = log_b(x) + log_b(y), and the power property states that log_b(x^n) = n * log_b(x). These properties are crucial for manipulating logarithmic expressions and solving logarithmic equations.

Change of Base Formula

The change of base formula allows you to convert logarithms from one base to another, expressed as log_b(a) = log_k(a) / log_k(b) for any positive k. This is particularly useful when dealing with logarithms of bases that are not easily computable, enabling the evaluation of logarithmic expressions in a more manageable form.