Domanda del libro di testo
Graph functions f and g in the same rectangular coordinate system. Graph and give equations of all asymptotes. If applicable, use a graphing utility to confirm your hand-drawn graphs. f(x) = 3x and g(x) = 3-x
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- 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
Capitolo 5, Problema 45
Use properties of logarithms to condense each logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. log2 (96) - log2 (3)
Guida verificata passo dopo passo1
Identify the logarithmic expression given: \(\log_{2}(96) - \log_{2}(3)\).
Recall the logarithmic property for subtraction: \(\log_{a}(x) - \log_{a}(y) = \log_{a}\left(\frac{x}{y}\right)\), where \(a\), \(x\), and \(y\) are positive and \(a \neq 1\).
Apply this property to combine the expression into a single logarithm: \(\log_{2}\left(\frac{96}{3}\right)\).
Simplify the fraction inside the logarithm: calculate \(\frac{96}{3}\) to get a simpler argument for the logarithm.
Rewrite the expression as \(\log_{2}(\text{simplified value})\), which is the condensed form with coefficient 1.
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Properties of Logarithms
Properties of logarithms include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions. For example, the quotient rule states that log_b(A) - log_b(B) = log_b(A/B), which is essential for condensing expressions.
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5:36
Change of Base Property
Logarithmic Expression Simplification
Simplifying logarithmic expressions involves rewriting them as a single logarithm with coefficient 1. This often requires applying properties of logarithms and factoring numbers inside the log to simplify or evaluate without a calculator.
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7:30Logarithms Introduction
Evaluating Logarithms Without a Calculator
Evaluating logarithms without a calculator involves expressing numbers as powers of the base or factoring to simplify. For example, recognizing that 96/3 = 32, and since 32 = 2^5, log_2(32) = 5, allows exact evaluation.
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5:14Evaluate Logarithms
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The figure shows the graph of f(x) = ex. In Exercises 35-46, use transformations of this graph to graph each function. Be sure to give equations of the asymptotes. Use the graphs to determine graphs. each function's domain and range. If applicable, use a graphing utility to confirm your hand-drawn h(x) = e2x + 1
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