Select the correct choice that completes the sentence below. $$ y=\ln x $$ if and only if $$ \square $$ \[ \begin{array}{l} x=10^{y} \\ x=e^{y} \\ y=10^{x} \\ y=e^{x} \end{array} \]

Given $$ y = \ln x $$, by the definition of the natural logarithm, the inverse function is given by \[ x = e^y. \] Thus, the correct completion of the sentence is: \[ x = e^y. \] $$ y = \ln x $$ if and only if $$ x = e^{y} $$.

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Question

Select the correct choice that completes the sentence below.
if and only if

Ask by Dickson Riley. in the United States

Mar 26,2025

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Answer

if and only if

Solution

Given

, by the definition of the natural logarithm, the inverse function is given by

Thus, the correct completion of the sentence is:

Answered by UpStudy AI and reviewed by a Professional Tutor

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Bonus Knowledge

The correct choice that completes the sentence is

When you have the natural logarithm

, it indicates that

is equal to

raised to the power of

. This is a fundamental property of logarithms and exponents, connecting the natural logarithm and the base of the natural logarithm,

Now, let’s dive into some fun facts about logarithms! Did you know that the concept of logarithms was first introduced by John Napier in the early 17th century? They were initially developed to simplify complex calculations, especially for astronomers and navigators. Today, they’re used in everything from computer algorithms to measuring the intensity of earthquakes!

Also, if you’re curious about how logs fit into our daily lives, take a moment to consider the decibel scale used for sound measurement. The decibel, which quantifies sound intensity, is a logarithmic scale, meaning a sound measured at 60 decibels is actually 10 times more intense than one measured at 50 decibels. So next time you crank up your favorite playlist, remember—you’re diving into the world of logarithms!

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Select the correct choice that completes the sentence below. if and only if