Table of contents

0. Functions7h 52m
1. Limits and Continuity2h 2m
2. Intro to Derivatives1h 33m
3. Techniques of Differentiation3h 18m
4. Applications of Derivatives2h 38m
5. Graphical Applications of Derivatives6h 2m
6. Derivatives of Inverse, Exponential, & Logarithmic Functions2h 37m
7. Antiderivatives & Indefinite Integrals1h 26m
8. Definite Integrals4h 44m
9. Graphical Applications of Integrals2h 27m
- Area Between Curves
  1h 18m
- Introduction to Volume & Disk Method
  1h 8m
10. Physics Applications of Integrals 3h 16m
- Kinematics
  1h 13m
- Work
  2h 3m
11. Integrals of Inverse, Exponential, & Logarithmic Functions2h 34m
12. Techniques of Integration7h 39m
13. Intro to Differential Equations2h 55m
14. Sequences & Series5h 36m
15. Power Series2h 19m
- Introduction to Power Series
  1h 9m
- Taylor Series & Taylor Polynomials
  1h 10m
16. Parametric Equations & Polar Coordinates7h 58m

0. Functions

Logarithmic Functions

Calculus

0. Functions

Logarithmic Functions

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2:38 minutes

Problem 7.1.4

Textbook Question

What is the inverse function of ln x, and what are its domain and range?

Verified step by step guidance

The inverse function of ln(x) is the exponential function, denoted as e^x. This is because the natural logarithm ln(x) and the exponential function e^x are inverse operations of each other.

To find the inverse function, start by setting y = ln(x). Then rewrite the equation in exponential form: x = e^y. This shows that the inverse function of ln(x) is f^(-1)(x) = e^x.

The domain of the natural logarithm function ln(x) is (0, ∞), meaning it is defined for all positive real numbers. Therefore, the range of its inverse function, e^x, is also (0, ∞).

The range of the natural logarithm function ln(x) is (-∞, ∞), meaning it can output any real number. Therefore, the domain of its inverse function, e^x, is also (-∞, ∞).

In summary, the inverse function of ln(x) is f^(-1)(x) = e^x, with a domain of (-∞, ∞) and a range of (0, ∞).

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Key Concepts

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

Inverse Functions

An inverse function essentially reverses the effect of the original function. For a function f(x), its inverse f⁻¹(x) satisfies the condition f(f⁻¹(x)) = x for all x in the domain of f⁻¹. Understanding how to find the inverse involves swapping the roles of the input and output and solving for the new output.

Natural Logarithm (ln x)

The natural logarithm, denoted as ln x, is the logarithm to the base e, where e is approximately 2.718. It is defined for positive real numbers, and its output is the exponent to which e must be raised to obtain x. The function ln x is crucial in calculus for solving exponential equations and understanding growth processes.

Domain and Range

The domain of a function is the set of all possible input values (x-values) for which the function is defined, while the range is the set of all possible output values (y-values). For the natural logarithm ln x, the domain is (0, ∞) since it is only defined for positive x, and the range is (-∞, ∞) because ln x can take any real number as its output.