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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?
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00:59
Clarissa Noh
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
Lectures
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A multivariate function is a function whose value depends on several variables. In contrast, a univariate function is a function whose value depends on only one variable. A multivariate function is also called a multivariate expression, a multivariate polynomial, a multivariate series, or a multivariate function of several variables.
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In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.
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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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