Textbook Question
Graph each function by making a table of coordinates. If applicable, use a graphing utility to confirm your hand-drawn graph. f(x) = (0.6)x
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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 17
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
Verified step by step guidance1
Recognize that the expression is \( \log N^{-6} \), which involves a logarithm of a power.
Use the logarithmic property that \( \log a^b = b \log a \) to rewrite the expression.
Apply this property to get \( \log N^{-6} = -6 \log N \).
Since the expression is now expanded, check if \( \log N \) can be simplified further or evaluated without a calculator (usually it cannot unless \(N\) is a known value).
Conclude that the fully expanded form of the expression is \( -6 \log N \).
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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(a^b) = b * log(a), which is essential for expanding expressions with exponents.
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Negative Exponents in Logarithms
A negative exponent indicates the reciprocal of the base raised to the positive exponent, i.e., a^{-b} = 1/a^b. When applied inside a logarithm, this affects the expression by introducing a negative sign outside the logarithm, as log(a^{-b}) = -b * log(a). Understanding this helps in correctly expanding and simplifying the expression.
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Evaluating Logarithms Without a Calculator
Evaluating logarithms without a calculator involves recognizing logarithms of numbers that are powers of the base or using known logarithmic values. Simplifying expressions using properties can reduce complex logs to simpler forms or constants, making it possible to evaluate them exactly without computational tools.
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Related Practice
Textbook Question
Write each equation in its equivalent logarithmic form. b3 = 1000
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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. 4x=1/√2
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Textbook Question
In Exercises 16–18, write each equation in its equivalent logarithmic form. 13^y = 874
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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
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