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Graph the function $ f(x) = \sqrt{x^3 + x^2 + x +…

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Problem 58 Hard Difficulty

(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?

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Video Transcript

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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Graph the function $ f(x) = \sqrt{x^3 + x^2 + x +…

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James Stewart

Calculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Kayleah Tsai

Caleb Elmore

This textbook answer is only visible when subscribed! Please subscribe to view the answer

Calculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Kayleah Tsai

Caleb Elmore

James StewartCalculus: Early Transcendentals

Catherine Ross

Anna Marie Vagnozzi

Kayleah Tsai

Caleb Elmore

James Stewart

Calculus: Early Transcendentals