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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook
All textbooksBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblem 45
Chapter 5, Problem 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)

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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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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 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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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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Logarithms 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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Related Practice
Textbook Question

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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Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. 32x+3x−2=0

Textbook Question

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) = (1/3). 3x

Textbook Question

Graph f(x) = (1/2)x and g(x) = log1/2 x in the same rectangular coordinate system.

Textbook Question

Solve each exponential equation in Exercises 23–48. Express the solution set in terms of natural logarithms or common logarithms. Then use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. e4x+5e2x−24=0

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Textbook Question

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