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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 92
Chapter 5, Problem 92

Determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. ln(8x3) = 3 ln (2x)

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Recall the logarithmic property that states \(\ln(a^b) = b \ln(a)\), which allows us to rewrite logarithms of powers.
Rewrite the left side \(\ln(8x^3)\) by expressing 8 as \$2^3$, so it becomes \(\ln(2^3 \cdot x^3)\).
Use the logarithm product rule: \(\ln(ab) = \ln(a) + \ln(b)\), to separate \(\ln(2^3 \cdot x^3)\) into \(\ln(2^3) + \ln(x^3)\).
Apply the power rule to each term: \(\ln(2^3) = 3 \ln(2)\) and \(\ln(x^3) = 3 \ln(x)\), so the left side becomes \(3 \ln(2) + 3 \ln(x)\).
Rewrite the right side \(3 \ln(2x)\) using the product rule: \(3 (\ln(2) + \ln(x)) = 3 \ln(2) + 3 \ln(x)\), and compare it to the left side to determine if the equation is true.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Properties of Logarithms

Logarithmic properties, such as the product, quotient, and power rules, allow us to simplify and manipulate logarithmic expressions. For example, ln(ab) = ln(a) + ln(b) and ln(a^n) = n ln(a). These rules are essential for breaking down and comparing logarithmic equations.
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Change of Base Property

Equivalence of Logarithmic Expressions

Two logarithmic expressions are equal if and only if their arguments are equal, assuming the logarithm base is the same and the arguments are within the domain. This concept helps verify if an equation like ln(8x^3) = 3 ln(2x) holds true by comparing the expressions inside the logarithms.
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Logarithms Introduction

Algebraic Manipulation of Exponents and Factors

Understanding how to factor and expand expressions with exponents is crucial. For instance, recognizing that 8x^3 = (2^3)(x^3) and that 3 ln(2x) = 3(ln 2 + ln x) helps in rewriting and comparing both sides of the equation accurately.
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Related Practice
Textbook Question

Graph f(x) = 2x and its inverse function in the same rectangular coordinate system.

Textbook Question

n Exercises 92–93, rewrite the equation in terms of base e. Express the answer in terms of a natural logarithm and then round to three decimal places. y = 73(2.6)^x

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Textbook Question

In Exercises 89–102, determine whether each equation is true or false. Where possible, show work to support your conclusion. If the statement is false, make the necessary change(s) to produce a true statement. log4 (2x3) = 3 log4 (2x)

Textbook Question

Evaluate or simplify each expression without using a calculator. In (1/e6)

Textbook Question

Evaluate or simplify each expression without using a calculator. eln 125

Textbook Question

Solve each logarithmic equation in Exercises 49–92. Be sure to reject any value of x that is not in the domain of the original logarithmic expressions. Give the exact answer. Then, where necessary, use a calculator to obtain a decimal approximation, correct to two decimal places, for the solution. ln(x−2)−ln(x+3)=ln(x−1)−ln(x+7)

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