Skip to main content
Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.R.1c
Chapter 7, Problem 7.R.1c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
c. ln xy = (ln x)(ln y)

Verified step by step guidance
1
Recall the logarithm property for the product of two positive numbers: \(\ln(xy) = \ln x + \ln y\). This is a fundamental identity in logarithms.
Compare the given statement \(\ln xy = (\ln x)(\ln y)\) with the known property. The statement suggests that the logarithm of a product equals the product of the logarithms, which differs from the sum in the known property.
To test the statement, consider specific positive values for \(x\) and \(y\), for example, \(x=2\) and \(y=3\). Calculate both sides: \(\ln(2 \times 3)\) and \((\ln 2)(\ln 3)\) to see if they are equal.
Since \(\ln(6)\) is approximately \(1.79\), and \((\ln 2)(\ln 3)\) is approximately \(0.48\), the two sides are not equal, providing a counterexample.
Conclude that the statement \(\ln xy = (\ln x)(\ln y)\) is false because the logarithm of a product is the sum of the logarithms, not their product.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
1m
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 simplify expressions, such as ln(xy) = ln(x) + ln(y). Understanding these properties helps determine the validity of logarithmic equations.
Recommended video:
05:36
Change of Base Property

Difference Between Addition and Multiplication

In logarithmic identities, multiplication inside the log translates to addition outside, not multiplication. Recognizing this distinction is crucial to avoid incorrect assumptions like ln(xy) = (ln x)(ln y).
Recommended video:
02:59
Finding Area Between Curves that Cross on the Interval

Counterexamples in Mathematical Proofs

A single counterexample disproves a general statement. Testing the equation with specific values of x and y can show whether ln(xy) = (ln x)(ln y) holds true or not.
Recommended video:
05:13
Slopes of Tangent Lines
Related Practice
Textbook Question

Surface area of a catenoid When the catenary y = a cosh x/a is revolved about the x-axis, it sweeps out a surface of revolution called a catenoid. Find the area of the surface generated when y = cosh x on [–ln 2, ln 2] is rotated about the x-axis.

120
views
Textbook Question

63–66. Calculator limits Use a calculator to make a table similar to Table 7.1 to approximate the following limits. Confirm your result with l’Hôpital’s Rule.


limₕ→₀ (1 + 3h)^{2/h}

31
views
Textbook Question

10–19. Derivatives Find the derivatives of the following functions.


f(t) = cosh t sinh t

34
views
Textbook Question

27–28. Curve sketching Use the graphing techniques of Section 4.4 to graph the following functions on their domains. Identify local extreme points, inflection points, concavity, and end behavior. Use a graphing utility only to check your work.

f(x) = ln x – ln² x

69
views
Textbook Question

22–36. Derivatives Find the derivatives of the following functions.


f(x) = x sinh⁻¹ x − √(x² + 1)

94
views
Textbook Question

10–19. Derivatives Find the derivatives of the following functions.

f(x) = tanh⁻¹(cos x)

28
views