Textbook Question
133. What is the age of a sample of charcoal in which 90% of the carbon-14 originally present has decayed?
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.PE.81
In Exercises 79–84, solve for y.
81. 9e^(2y) = = x^2
Verified step by step guidance1
Start with the given equation: \(9e^{2y} = x^2\).
Isolate the exponential term by dividing both sides of the equation by 9: \(e^{2y} = \frac{x^2}{9}\).
To solve for \(y\), take the natural logarithm (ln) of both sides to undo the exponential: \(\ln\left(e^{2y}\right) = \ln\left(\frac{x^2}{9}\right)\).
Use the logarithm property \(\ln\left(e^a\right) = a\) to simplify the left side: \(2y = \ln\left(\frac{x^2}{9}\right)\).
Finally, solve for \(y\) by dividing both sides by 2: \(y = \frac{1}{2} \ln\left(\frac{x^2}{9}\right)\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Exponential Functions
An exponential function involves a variable in the exponent, such as e^(2y). Understanding how to manipulate and isolate the variable in the exponent is essential for solving equations involving exponential expressions.
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Exponential Functions
Logarithmic Functions and Properties
Logarithms are the inverse operations of exponentials. Applying logarithms allows us to solve for variables in the exponent by converting exponential equations into linear ones, making it easier to isolate and solve for y.
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Algebraic Manipulation
Solving for y requires careful algebraic steps, including isolating terms, simplifying expressions, and applying inverse operations. Mastery of algebraic manipulation ensures accurate and efficient problem-solving.
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Related Practice
Textbook Question
Evaluate the integrals in Exercises 31–78.
64. ∫(from 1 to e)(8ln3 log_3(θ))/θ dθ
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Textbook Question
In Exercises 129–132 solve the initial value problem.
129. dy/dx = e^(-x-y-2), y(0) = -2
Textbook Question
Use l’Hôpital’s Rule to find the limits in Exercises 85–108.
91. lim(x→π/2⁻) (sec(7x))(cos(3x))
Textbook Question
109. Does f grow faster, slower, or at the same rate as g as x→∞? Give reasons for your answers.
e. f(x) = arccsc(x), g(x) = 1/x
Textbook Question
In Exercises 79–84, solve for y.
83. ln(y-1) = x + ln(y)