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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1:49 minutes

Problem 23

Textbook Question

Solve each equation. x = log￬8 ∜8

Verified step by step guidance

Recognize that the equation given is \( x = \log_8 \sqrt[4]{8} \).

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

Express \( \sqrt[4]{8} \) as an exponent: \( 8^{1/4} \).

Apply the logarithm property: \( x = \log_8 (8^{1/4}) = \frac{1}{4} \cdot \log_8 8 \).

Use the fact that \( \log_b b = 1 \) to simplify: \( \log_8 8 = 1 \), so \( x = \frac{1}{4} \cdot 1 \).

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Key Concepts

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

Logarithms

Logarithms are the inverse operations of exponentiation. They answer the question: to what exponent must a base be raised to produce a given number? For example, in the expression log_b(a) = c, b^c = a. Understanding logarithms is essential for solving equations involving them, as they allow us to manipulate exponential relationships.

Change of Base Formula

The Change of Base Formula allows us 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 concept is crucial for solving logarithmic equations when the base is not easily computable.

Exponential Equations

Exponential equations involve variables in the exponent and can often be solved using logarithms. For instance, if we have an equation like a^x = b, we can take the logarithm of both sides to isolate x. Understanding how to manipulate and solve exponential equations is key to finding solutions in problems involving logarithms.