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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2:18 minutes

Problem 107

Textbook Question

In Exercises 105–108, evaluate each expression without using a calculator. log2 (log3 81)

Verified step by step guidance

Identify the innermost logarithm: \( \log_3 81 \).

Recognize that 81 is a power of 3: \( 81 = 3^4 \).

Apply the logarithm power rule: \( \log_b (a^n) = n \cdot \log_b a \).

Calculate \( \log_3 81 = \log_3 (3^4) = 4 \cdot \log_3 3 \).

Since \( \log_3 3 = 1 \), we have \( \log_3 81 = 4 \). Now evaluate \( \log_2 4 \).

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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, allowing us to solve for the exponent in equations of the form b^y = x, where b is the base. The logarithm log_b(x) answers the question: 'To what power must b be raised to obtain x?' Understanding logarithms is essential for evaluating expressions involving them.

Change of Base Formula

The Change of Base Formula allows us to convert logarithms from one base to another, which is particularly useful when dealing with bases that are not easily computable. The formula states that log_b(a) = log_k(a) / log_k(b) for any positive k. This concept is crucial for simplifying logarithmic expressions, especially when evaluating nested logarithms.

Evaluating Logarithmic Expressions

Evaluating logarithmic expressions involves determining the value of the logarithm based on known values or properties. For example, log_3(81) can be simplified by recognizing that 81 is 3 raised to the power of 4, leading to log_3(3^4) = 4. Mastery of this concept is necessary for solving complex logarithmic problems without a calculator.