Textbook Question
In Exercises 13–15, write each equation in its equivalent exponential form. 1/2 = log49 7
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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
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb x3
Verified step by step guidance1
Identify the logarithmic expression given: \(\log_{b} x^{3}\).
Recall the logarithmic power rule: \(\log_{b} (a^{c}) = c \cdot \log_{b} a\).
Apply the power rule to the expression: \(\log_{b} x^{3} = 3 \cdot \log_{b} x\).
Since \(\log_{b} x\) cannot be simplified further without additional information, the expanded form is \(3 \log_{b} x\).
Thus, the expression is fully expanded using properties of logarithms.
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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 us to rewrite logarithmic expressions in simpler or expanded forms. For example, the power rule states that log_b(x^n) = n * log_b(x), which is essential for expanding expressions like log_b(x^3).
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Change of Base Property
Logarithmic Functions and Notation
A logarithm log_b(x) answers the question: to what power must the base b be raised to get x? Understanding this definition helps in manipulating and interpreting logarithmic expressions. The base b must be positive and not equal to 1, and x must be positive for the logarithm to be defined.
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Evaluating Logarithms Without a Calculator
Some logarithmic expressions can be simplified or evaluated exactly using known values or properties, such as log_b(b) = 1 or log_b(1) = 0. Recognizing when an expression can be rewritten to use these values helps avoid calculator use and deepens understanding of logarithmic behavior.
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Related Practice
Textbook Question
Write each equation in its equivalent exponential form. log3 81 = y
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. 4x=1/√2
Textbook Question
Write each equation in its equivalent logarithmic form. 132 = x
Textbook Question
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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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. 6(x−3)/4=√6