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Inverse of the Exponential Function

publish date: 2026/05/23 21:16:57.802394 UTC

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The exponential function \(y = e^x\) defines a bijective mapping from \((-\infty, \infty)\) onto \((0, \infty)\). What is its inverse function, and what are the inverse's domain and range?

Inverse: \(y = e^{-x}\); domain \((0, \infty)\); range \((-\infty, \infty)\)

Inverse: \(y = \ln x\); domain \((0, \infty)\); range \((-\infty, \infty)\)

Inverse: \(y = \ln x\); domain \((-\infty, \infty)\); range \((0, \infty)\)

Inverse: \(y = \log_{10} x\); domain \((-\infty, \infty)\); range \((0, \infty)\)

Correct Answer

Inverse: \(y = \ln x\); domain \((0, \infty)\); range \((-\infty, \infty)\)

Explanation

From \(y = e^x\): \(x = \ln y\). Replacing \(y\) with \(x\): \(y = \ln x\). The domain of \(\ln x\) is the range of \(e^x\), which is \((0,\infty)\), and its range is \((-\infty,\infty)\). This illustrates a surprising fact: a one-to-one correspondence can exist between intervals of different apparent sizes.

Reference

Introduction to Differential Calculus (Systematic Studies with Engineering Applications for Beginners) - 2012

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