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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblema 7.1.4
Capitolo 7, Problema 7.1.4

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

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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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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.
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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.
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Derivative of the Natural Logarithmic Function

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.
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Finding the Domain and Range of a Graph
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