Textbook Question
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb x3
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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 15
Write each equation in its equivalent logarithmic form. 132 = x
Verified step by step guidance1
Identify the given exponential equation: \(13^2 = x\).
Recall the definition of logarithms: if \(a^b = c\), then the equivalent logarithmic form is \(\log_a c = b\).
In this problem, the base \(a\) is 13, the exponent \(b\) is 2, and the result \(c\) is \(x\).
Apply the logarithmic form to rewrite the equation as \(\log_{13} x = 2\).
This is the equivalent logarithmic form of the given exponential equation.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential and Logarithmic Forms
Exponential and logarithmic forms are two ways to express the same relationship. An equation like a^b = c can be rewritten as log_a(c) = b, where the logarithm answers the question: to what power must the base a be raised to get c?
Recommended video:
5:02
Solving Logarithmic Equations
Definition of a Logarithm
A logarithm log_a(x) is the inverse operation of exponentiation, meaning it finds the exponent to which the base a must be raised to produce x. This definition is fundamental for converting between exponential and logarithmic equations.
Recommended video:
7:30Logarithms Introduction
Properties of Logarithms
Understanding properties such as the base must be positive and not equal to 1, and the argument must be positive, is essential. These properties ensure the logarithmic form is valid and help in correctly rewriting exponential equations.
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5:36Change of Base Property
Related Practice
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Write each equation in its equivalent exponential form. log3 81 = y
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Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log N-6
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Write each equation in its equivalent logarithmic form. b3 = 1000
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Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. h(x) = (1/2)x
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Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 6(x−3)/4=√6
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