Textbook Question
Write each equation in its equivalent logarithmic form. 2-4 = 1/16
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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 11
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4 (64/y)
Verified step by step guidance1
Recall the logarithmic property for the logarithm of a quotient: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\).
Apply this property to the given expression: \(\log_4 \left( \frac{64}{y} \right) = \log_4 64 - \log_4 y\).
Recognize that 64 is a power of 4 since \(64 = 4^3\).
Use the logarithmic identity \(\log_b (b^k) = k\) to simplify \(\log_4 64\) as \(\log_4 (4^3) = 3\).
Write the fully expanded expression as \(3 - \log_4 y\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms
Properties of logarithms include rules such as the product, quotient, and power rules. These allow you to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N), which is essential for breaking down complex expressions.
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Change of Base Property
Change of Base and Evaluating Logarithms
Evaluating logarithms often involves expressing numbers as powers of the base. For instance, 64 can be written as 4^3, so log_4(64) = 3. Recognizing these relationships helps simplify expressions without a calculator.
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Simplifying Algebraic Expressions
Simplifying expressions like log_4(64/y) requires understanding how to separate terms using logarithm properties and handle variables correctly. This involves rewriting the expression as log_4(64) - log_4(y) and simplifying each part individually.
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05:07Simplifying Algebraic Expressions
Related Practice
Textbook Question
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 9x=27
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Textbook Question
In Exercises 11–18, graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. f(x) = 4x
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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. ln(e2/5)
Textbook Question
Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. g(x) = (3/2)x
Textbook Question
Write each equation in its equivalent logarithmic form. 54 = 625
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