Ch. 7 - Transcendental Functions

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 7 - Transcendental Functions

Problem 7.2.1d

Chapter 7, Problem 7.2.1d

1. Express the following logarithms in terms of ln 2 and ln 3.
d. ln ∛9

Verified step by step guidance

Recall that the cube root of 9 can be expressed as an exponent: \(\sqrt[3]{9} = 9^{\frac{1}{3}}\).

Rewrite the logarithm using the exponent rule: \(\ln \sqrt[3]{9} = \ln \left(9^{\frac{1}{3}}\right) = \frac{1}{3} \ln 9\).

Express 9 as a power of 3: \(9 = 3^2\).

Use the logarithm power rule to rewrite \(\ln 9\): \(\ln 9 = \ln \left(3^2\right) = 2 \ln 3\).

Substitute back to get the expression in terms of \(\ln 3\): \(\ln \sqrt[3]{9} = \frac{1}{3} \times 2 \ln 3 = \frac{2}{3} \ln 3\).

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Key Concepts

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

Properties of Logarithms

Logarithmic properties such as the power rule (ln(a^b) = b ln(a)) and product rule allow us to rewrite complex logarithms in simpler forms. These rules help express logarithms of roots and products in terms of basic logarithms.

Natural Logarithm (ln)

The natural logarithm, denoted ln, is the logarithm with base e. Understanding ln and its behavior is essential for manipulating expressions involving logarithms, especially when converting between different bases or simplifying expressions.

Expressing Numbers as Powers or Products

To rewrite logarithms in terms of ln 2 and ln 3, numbers like 9 must be expressed as powers or products of 2 and 3. For example, 9 = 3^2, which allows the use of logarithm properties to break down ln(∛9) into terms involving ln 3.

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Textbook Question

2. Express the following logarithms in terms of ln 5 and ln 7.

d. ln 1225

Textbook Question

86. This exercise explores the difference between

lim(x→∞)(1 + 1/x²)^x

and

lim(x→∞)(1 + 1/x)^x = e

c. Confirm your estimate of lim(x→∞)f(x) by calculating it with l’Hôpital’s Rule.

Textbook Question

In Exercises 67–72, you will explore some functions and their inverses together with their derivatives and tangent line approximations at specified points. Perform the following steps using your CAS:

d. Find the equation for the tangent line to g at the point (f(x_0), x_0) located symmetrically across the 45° line y=x (which is the graph of the identity function). Use Theorem 1 to find the slope of this tangent line.

68. y= (3x+2)/(2x-11), -2 ≤ x ≤ 2, x_0=1/2

Textbook Question

155. Which is bigger, πᵉ or e^π?

Calculators have taken some of the mystery out of this once-challenging question.

(Go ahead and check; you will see that it is a very close call.)

You can answer the question without a calculator, though.

d. Conclude that

xᵉ < eˣfor all positivex ≠ e.

Textbook Question

70. y= x³/(x²+1), -1 ≤ x ≤ 1, x_0=1/2

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Textbook Question