Skip to main content
Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 7 - Transcendental FunctionsProblem 7.2.4a
Chapter 7, Problem 7.2.4a

4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
a. ln secθ + ln cosθ

Verified step by step guidance
1
Recall the logarithm property that states: \(\ln a + \ln b = \ln (a \times b)\). This means the sum of two logarithms with the same base can be combined into the logarithm of the product of their arguments.
Identify the expressions inside the logarithms: here, we have \(\ln \sec \theta\) and \(\ln \cos \theta\).
Apply the property by multiplying the arguments inside the logarithms: \(\ln (\sec \theta \times \cos \theta)\).
Simplify the product inside the logarithm. Recall that \(\sec \theta = \frac{1}{\cos \theta}\), so multiply \(\sec \theta\) and \(\cos \theta\) accordingly.
Write the final expression as a single logarithm term: \(\ln (\sec \theta \cdot \cos \theta)\), which simplifies to \(\ln (1)\).

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

Logarithmic properties include rules such as the product, quotient, and power rules. The product rule states that the sum of logarithms with the same base can be combined into the logarithm of the product of their arguments, i.e., ln(a) + ln(b) = ln(ab). This is essential for rewriting expressions as a single logarithmic term.
Recommended video:
05:36
Change of Base Property

Trigonometric Functions and Identities

Understanding basic trigonometric functions like secant (sec θ = 1/cos θ) and cosine is crucial. Recognizing these relationships allows simplification of expressions inside logarithms, which helps in combining or reducing terms effectively.
Recommended video:
6:04
Introduction to Trigonometric Functions

Simplification of Logarithmic Expressions

Simplifying logarithmic expressions involves applying logarithm properties and algebraic manipulation to rewrite complex expressions into simpler or single terms. This skill is important for solving problems efficiently and for clearer interpretation of results.
Recommended video:
7:30
Logarithms Introduction
Related Practice
Textbook Question

Evaluate the integrals in Exercises 31–78.

45. ∫(ln x)^(-3)/x dx

23
views
Textbook Question

80. Find all values of c that satisfy the conclusion of Cauchy's Mean Value Theorem for the given functions and interval.

a. f(x) = x, g(x) = x², (a, b) = (-2, 0)

18
views
Textbook Question

In Exercises 41–44:

a. Find f⁻¹(x).


43. f(x) = 5 − 4x, a = 1/2

23
views
Textbook Question

5. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?

a. log_3(x)

29
views
Textbook Question

In Exercises 1–24, find the derivative of y with respect to the appropriate variable.

1. y = 10e^(-x/5)

22
views
Textbook Question

23. What roles do the functions e^x and ln(x) play in growth comparisons?

20
views