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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1:41 minutes

Problem 31

Textbook Question

In Exercises 21–42, evaluate each expression without using a calculator. log2 (1/√2)

Verified step by step guidance

Recognize that the expression is a logarithm with base 2: \( \log_2 \left( \frac{1}{\sqrt{2}} \right) \).

Rewrite \( \frac{1}{\sqrt{2}} \) as \( 2^{-1/2} \) because \( \sqrt{2} = 2^{1/2} \) and \( \frac{1}{\sqrt{2}} = 2^{-1/2} \).

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

Substitute \( a = 2 \), \( c = -1/2 \), and \( b = 2 \) into the power rule: \( \log_2 (2^{-1/2}) = -\frac{1}{2} \cdot \log_2 (2) \).

Since \( \log_2 (2) = 1 \), simplify the expression to find the result.

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Key Concepts

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

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. The logarithm log_b(a) answers the question: 'To what power must the base b be raised to obtain a?' Understanding this concept is crucial for evaluating logarithmic expressions, as it allows us to manipulate and simplify them effectively.

Properties of Logarithms

Properties of logarithms, such as the product, quotient, and power rules, provide essential tools for simplifying logarithmic expressions. For instance, the property log_b(m/n) = log_b(m) - log_b(n) allows us to break down complex logarithmic expressions into simpler components, facilitating easier evaluation.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which can be particularly useful when dealing with logarithms of bases that are not easily computable. The formula states that log_b(a) = log_k(a) / log_k(b) for any positive k, enabling us to evaluate logarithms using more familiar bases.