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(a) What are the values of $ e^{\ln 300} $ and $ \ln (e^{300}) $?(b) Use your calculator to evaluate $ e^{\ln 300} $ and $ \ln (e^{300}) $. What do you notice? Can you explain why the calculator has trouble?

a) $e^{\ln 300}=300$ and $\ln e^{300}=300$b) See explanation

00:59

Clarissa N.

Calculus 1 / AB

Calculus 2 / BC

Calculus 3

Chapter 1

Functions and Models

Section 5

Inverse Functions and Logarithms

Functions

Integration Techniques

Partial Derivatives

Functions of Several Variables

Missouri State University

Campbell University

Harvey Mudd College

Baylor University

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to find the value of these two expressions. We want to think about the fact that the natural log function and the E to the X function are in verses of one another and because there, in verses of one another, when you compose them, put one inside the other, they're going to cancel. And so if you had e to the natural log of X, it would just be X. And if you have the natural log of each of the X, it would just be X, because the inverse functions cancel one another. So if you have E to the natural log of 300 is just going to be 300. Or if you have the natural log of each of the 300 it's going to be 300. Now you can try putting these into your calculator, and for my calculator, it was actually able to do the 1st 1 just fine. I typed any to the natural log of 300. It gave me 300. No problem. However, when I typed in the other one, it gave me overflow error. So why did that happen? So chances are the calculator was attempting to calculate each of the 3/100 power before taking the natural log. It wasn't thinking about the concept. It only knows, um, operations and number crunching. And so it wasn't thinking about the concept of inverse functions and an attempt to find each of the 300 power that was just too big. Sometimes you just have to be smarter than your calculator.

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