Textbook Question
Graph f(x) = 4x and g(x) = log4 x in the same rectangular coordinate system.
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 43
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. ln x + ln 7
Verified step by step guidance1
Recall the logarithmic property that states the sum of logarithms with the same base can be written as the logarithm of the product: \(\ln a + \ln b = \ln(ab)\).
Identify the terms in the expression: \(\ln x + \ln 7\); here, \(a = x\) and \(b = 7\).
Apply the property by combining the two logarithms into one: \(\ln(x) + \ln(7) = \ln(x \times 7)\).
Simplify the product inside the logarithm: \(\ln(7x)\).
The expression is now condensed into a single logarithm with coefficient 1: \(\ln(7x)\).
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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 product rule states that ln(a) + ln(b) = ln(ab), which helps condense sums of logarithms into a single logarithm.
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Change of Base Property
Natural Logarithm (ln)
The natural logarithm, denoted ln, is the logarithm with base e, where e ≈ 2.718. It is commonly used in calculus and algebra. Understanding ln is essential for manipulating expressions involving exponential growth or decay and applying logarithmic properties correctly.
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Condensing Logarithmic Expressions
Condensing logarithmic expressions means rewriting multiple logarithms as a single logarithm with a coefficient of 1. This involves applying logarithm properties to combine terms and simplify the expression, making it easier to evaluate or use in further calculations.
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Related Practice
Textbook Question
Evaluate each expression without using a calculator.
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. e2x−3ex+2=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 g(x) = 2ex
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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) = e-x
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. 5(2x+3)=3(x−1)