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 7.1.67c
Chapter 7, Problem 7.1.67c

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample. Assume x > 0 and y > 0.


c. ln (x + y) = ln x + ln y

Verified step by step guidance
1
Recall the logarithm property for multiplication: \( \ln(ab) = \ln a + \ln b \), which holds for positive \(a\) and \(b\).
Check if the given statement \( \ln(x + y) = \ln x + \ln y \) matches this property. Notice that the left side has \(x + y\) inside the logarithm, while the right side is a sum of logarithms.
Since the logarithm of a sum \( \ln(x + y) \) is not equal to the sum of logarithms \( \ln x + \ln y \), this suggests the statement is generally false.
To confirm, consider a counterexample: choose specific positive values for \(x\) and \(y\), such as \(x = 1\) and \(y = 1\), and evaluate both sides to see if they are equal.
Since the values will not be equal, conclude that \( \ln(x + y) \neq \ln x + \ln y \) in general, and the statement is false.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.
Video duration:
2m
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 relate sums and products. The key property is that the logarithm of a product equals the sum of the logarithms: ln(xy) = ln x + ln y. However, the logarithm of a sum, ln(x + y), does not equal the sum of logarithms, which is a common misconception.
Recommended video:
05:36
Change of Base Property

Counterexamples in Mathematical Proof

A counterexample is a specific case that disproves a general statement. To show that ln(x + y) ≠ ln x + ln y, one can choose positive values for x and y and demonstrate that the equality does not hold, thus proving the statement false.
Recommended video:
05:13
Slopes of Tangent Lines

Domain Restrictions for Logarithmic Functions

Logarithmic functions are defined only for positive arguments. Since x > 0 and y > 0, ln x, ln y, and ln(x + y) are all defined. Understanding the domain ensures the expressions are valid and helps avoid errors when evaluating or comparing logarithmic expressions.
Recommended video:
5:26
Graphs of Logarithmic Functions
Related Practice
Textbook Question

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

d. Plot a graph of V(t) for 0 ≤ t ≤ 15. What happens to the size of the tumor, assuming there are no follow-up treatments with Cisplatin?

40
views
Textbook Question

Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.


c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

43
views
Textbook Question

Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.

35
views
Textbook Question

Terminal velocity Refer to Exercises 95 and 96.


d. How tall must a cliff be so that the BASE jumper (m = 75 kg and k = 0.2) reaches 95% of terminal velocity? Assume the jumper needs at least 300 m at the end of free fall to deploy the chute and land safely.

70
views
Textbook Question

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.

100
views
Textbook Question

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


d. If the rate constant of an exponential growth function is increased, its doubling time is decreased.

53
views