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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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
Write each equation in its equivalent logarithmic form. 2-4 = 1/16
Verified step by step guidance1
Identify the components of the exponential equation \(2^{-4} = \frac{1}{16}\). Here, the base is 2, the exponent is -4, and the result is \(\frac{1}{16}\).
Recall the definition of logarithms: If \(a^x = b\), then the equivalent logarithmic form is \(\log_{a} b = x\).
Apply this definition to the given equation by setting \(a = 2\), \(b = \frac{1}{16}\), and \(x = -4\).
Write the logarithmic form as \(\log_{2} \left( \frac{1}{16} \right) = -4\).
This expresses the original exponential equation in its equivalent logarithmic form.
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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
An exponential equation like a^x = b can be rewritten in logarithmic form as log_a(b) = x. This conversion helps in solving for exponents by expressing the relationship between the base, exponent, and result in terms of logarithms.
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Properties of Exponents
Understanding how exponents work, including negative exponents, is essential. For example, a negative exponent indicates the reciprocal of the base raised to the positive exponent, such as 2^-4 = 1/(2^4) = 1/16.
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04:06Rational Exponents
Logarithm Definition and Notation
A logarithm log_a(b) answers the question: 'To what power must the base a be raised to get b?' Recognizing this definition allows one to rewrite exponential equations into logarithmic form accurately.
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Related Practice
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
Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 31-x=1/27
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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. log4 (64/y)
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