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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 60
Chapter 5, Problem 60

In Exercises 60–63, 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 x)(ln 1) = 0

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Step 1: Recall the property of the natural logarithm (ln) that states ln(1) = 0. This is because the natural logarithm of 1 is the exponent to which e must be raised to equal 1, and e^0 = 1.
Step 2: Substitute ln(1) = 0 into the given equation. The equation becomes (ln x)(0) = 0.
Step 3: Simplify the expression. Any number multiplied by 0 is 0, so the left-hand side simplifies to 0.
Step 4: Compare the simplified left-hand side (0) to the right-hand side (0). Since both sides are equal, the equation is true.
Step 5: Conclude that the given equation (ln x)(ln 1) = 0 is true, and no changes are necessary to make it true.

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Key Concepts

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

Natural Logarithm (ln)

The natural logarithm, denoted as ln, is the logarithm to the base e, where e is approximately 2.71828. It is a fundamental concept in algebra and calculus, often used to solve equations involving exponential growth or decay. Understanding the properties of logarithms, such as ln(1) = 0, is crucial for evaluating expressions involving ln.
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Properties of Logarithms

Logarithmic properties are rules that simplify the manipulation of logarithmic expressions. Key properties include the product rule, quotient rule, and power rule. For instance, the property ln(a) + ln(b) = ln(ab) helps in combining logarithms, while ln(1) = 0 is essential for evaluating expressions where the logarithm of one is involved.
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Evaluating Expressions

Evaluating expressions involves substituting values into mathematical formulas and simplifying them to determine their truth value. In this context, evaluating (ln x)(ln 1) requires understanding that ln(1) equals 0, which leads to the entire expression equating to 0. This concept is vital for determining the validity of the equation presented in the question.
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Related Practice
Textbook Question

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. h(x) = log x − 1

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Textbook Question

Graph y= 2x and x = 2y in the same rectangular coordinate system.

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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. log4(3x+2)=3

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Textbook Question

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. 3 ln x + 5 ln y - 6 ln z

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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. 5 ln(2x)=20

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Textbook Question

The figure shows the graph of f(x) = log x. In Exercises 59–64, use transformations of this graph to graph each function. Graph and give equations of the asymptotes. Use the graphs to determine each function's domain and range. g(x) = log(x − 1)

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