Ch. 3 - Derivatives

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 3 - Derivatives

Problem 3.9.87.b

Chapter 3, Problem 3.9.87.b

Explain why or why not. Determine whether the following statements are true and give an explanation or counterexample.

b. ln(x + 1) + ln(x − 1) = ln(x² − 1), for all x.

Verified step by step guidance

Step 1: Recall the logarithmic property that states \( \ln(a) + \ln(b) = \ln(ab) \). This property allows us to combine the logarithms on the left-hand side of the equation.

Step 2: Apply the property from Step 1 to the left-hand side: \( \ln(x + 1) + \ln(x - 1) = \ln((x + 1)(x - 1)) \).

Step 3: Simplify the expression \((x + 1)(x - 1)\) using the difference of squares formula: \((x + 1)(x - 1) = x^2 - 1\).

Step 4: Substitute the simplified expression from Step 3 back into the equation: \( \ln((x + 1)(x - 1)) = \ln(x^2 - 1) \).

Step 5: Conclude that the original statement \( \ln(x + 1) + \ln(x - 1) = \ln(x^2 - 1) \) is true for all \( x \) such that \( x > 1 \) or \( x < -1 \), because the domain of the logarithmic function requires that the arguments \( x + 1 \) and \( x - 1 \) are positive.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

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

Properties of Logarithms

Logarithms have specific properties that govern their behavior, such as the product, quotient, and power rules. The product rule states that ln(a) + ln(b) = ln(ab), while the quotient rule states that ln(a) - ln(b) = ln(a/b). Understanding these properties is essential for manipulating logarithmic expressions and verifying the validity of statements involving logarithms.

Domain of Logarithmic Functions

The domain of a logarithmic function is restricted to positive real numbers. For the expression ln(x + 1) + ln(x - 1), both x + 1 and x - 1 must be greater than zero, which implies x must be greater than 1. Recognizing the domain is crucial for determining the validity of logarithmic equations and ensuring that all terms are defined.

Counterexamples in Mathematics

A counterexample is a specific case that disproves a general statement. In the context of the given logarithmic equation, finding a value of x that makes the left-hand side unequal to the right-hand side serves as a counterexample. This method is vital in mathematical reasoning, as it helps to establish the truth or falsehood of statements by demonstrating exceptions.

Recommended video:

05:13

Slopes of Tangent Lines

Related Practice

Textbook Question

62–65. {Use of Tech} Graphing f and f'

b. Compute and graph f'.

f(x)=e^−x tan^−1 x on [0,∞)

158

views

Textbook Question

{Use of Tech} Power and energy The total energy in megawatt-hr (MWh) used by a town is given by E(t) = 400t+2400/π sin πt/12, where t≥0 is measured in hours, with t=0 corresponding to noon.

b. At what time of day is the rate of energy consumption a maximum? What is the power at that time of day?

283

views

Textbook Question

Use a graphing utility to graph the curve and the tangent line on the same set of axes.

y = (x + 5) / (x - 1); a = 3

343

views

Textbook Question

13-26 Implicit differentiation Carry out the following steps.

b. Find the slope of the curve at the given point.

sin y = 5x⁴−5; (1, π)