Textbook Question
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log √(100x)
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 5, Problem 31
Evaluate each expression without using a calculator. log2 (1/√2)
Verified step by step guidance1
Recall the logarithm property that allows you to rewrite the logarithm of a quotient or root: \(\log_b \left( \frac{1}{\sqrt{2}} \right) = \log_b (1) - \log_b (\sqrt{2})\).
Recognize that \(\sqrt{2}\) can be expressed as an exponent: \(\sqrt{2} = 2^{\frac{1}{2}}\).
Use the logarithm power rule: \(\log_b (a^c) = c \cdot \log_b (a)\), so \(\log_2 (\sqrt{2}) = \log_2 \left( 2^{\frac{1}{2}} \right) = \frac{1}{2} \cdot \log_2 (2)\).
Since \(\log_2 (2) = 1\), simplify the expression to \(\frac{1}{2} \cdot 1 = \frac{1}{2}\).
Recall that \(\log_2 (1) = 0\), so the original expression becomes \(0 - \frac{1}{2} = -\frac{1}{2}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Logarithm Definition
A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log_b(a) = c means b^c = a. Understanding this definition helps in rewriting and evaluating logarithmic expressions.
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Logarithms Introduction
Properties of Logarithms
Logarithms have key properties such as log_b(xy) = log_b(x) + log_b(y), log_b(x/y) = log_b(x) - log_b(y), and log_b(x^r) = r * log_b(x). These properties allow simplification of complex expressions without a calculator.
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Exponents and Radicals
Radicals can be expressed as fractional exponents, e.g., √2 = 2^(1/2). Recognizing this allows rewriting expressions like 1/√2 as 2^(-1/2), which simplifies the evaluation of logarithms by converting roots into powers.
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Related Practice
Textbook Question
Graph f(x) = 2^x and g(x) = log2 x in the same rectangular coordinate system. Use the graphs to determine each function's domain and range.
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Textbook Question
Begin by graphing f(x) = 2x. Then use transformations of this graph to graph the given function. Be sure to graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn graphs. g(x) = −2x
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Textbook Question
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log ∛(x/y)
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Textbook Question
Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 5ex=23
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Textbook Question
Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 3e5x=1977
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