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.
Ch. 7 - Transcendental Functions
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 7, Problem 7.3.1c
In Exercises 1–4, solve for t.
1. c. e^((ln 0.2)t) = 0.4
Verified step by step guidance1
Recognize that the equation is given as \(e^{(\ln 0.2) t} = 0.4\). The goal is to solve for \(t\).
Recall the property of exponents and logarithms: \(e^{\ln a} = a\). This means the expression \(e^{(\ln 0.2) t}\) can be rewritten as \((e^{\ln 0.2})^t = (0.2)^t\).
Rewrite the equation using this property: \((0.2)^t = 0.4\).
To solve for \(t\), take the natural logarithm of both sides: \(\ln((0.2)^t) = \ln(0.4)\).
Use the logarithm power rule to bring down the exponent: \(t \cdot \ln(0.2) = \ln(0.4)\). Then isolate \(t\) by dividing both sides by \(\ln(0.2)\): \(t = \frac{\ln(0.4)}{\ln(0.2)}\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Properties of Logarithms and Exponents
Understanding how logarithms and exponents interact is essential, especially that e^(ln a) = a. This allows simplification of expressions like e^((ln 0.2)t) to (0.2)^t, making the equation easier to solve.
Recommended video:
05:36
Change of Base Property
Solving Exponential Equations
Solving equations where the variable is in the exponent involves rewriting the equation in a comparable base or applying logarithms to isolate the variable. This process transforms the problem into a linear equation in terms of the variable.
Recommended video:
5:47Solving Exponential Equations Using Logs
Using Natural Logarithms to Isolate Variables
Applying the natural logarithm (ln) to both sides of an equation helps isolate the variable when it appears as an exponent. Since ln and e are inverse functions, this step simplifies the equation and allows solving for the unknown.
Recommended video:
05:18Derivative of the Natural Logarithmic Function
Related Practice
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
7. c. arcsec(-2)
Textbook Question
Use reference triangles in an appropriate quadrant to find the angles in Exercises 1–8.
3. c. sin^(-1)(-√3/2)
Textbook Question
4. Which of the following functions grow faster than x² as x→∞? Which grow at the same rate as x²? Which grow slower?
c. x²e^(-x)
Textbook Question
6. Which of the following functions grow faster than ln(x) as x→∞? Which grow at the same rate as ln(x)? Which grow slower?
c. 1/√x
Textbook Question
4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
c. 3ln ∛(t² - 1) - ln(t+1)