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# The Number e and the Natural Logarithm - Problem 2

###### Norm Prokup

###### Norm Prokup

**Cornell University**

PhD. in Mathematics

Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.

I want to talk about some properties of logs as they apply to the natural logarithm. If you remember, the definition of natural logarithm is; y equals lnx means x equals e to the y. And that means that y equals lnx and y equals e to the x of inverse functions. As a result of this inverse relationship, we have two identities.

First ln of e to the x equals x and second, e to the lnx equals x. Now the second identity is only true for x greater than 0 because that’s the domain of natural log of x. So this second one x greater than 0, but this one is true for all x.

Now let’s apply this on an example. The problem says to evaluate these four things lnv to the 4, that’s just a straight forward application of this identity here. lnv to the 4 would be 4. Ln of 1 over e³. We first have to write this as a single power of e. So that would be natural log of e to the -3. 1 over e³ is e to the -3 so lnv of e of the -3, is -3.

Same thing here I need to write this root e as a power of e, but that’s e to the 1/2. And so ln of e to the 1/2 is 1/2.

And finally and you may know the answer to this one but this is kind of e. If you express 1 as e to the 0, you see y the natural log of 1 is 0. Any log, log of any base of 1 is 0, but you can use this inverse property to show it in a kind of nice way.

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###### Norm Prokup

PhD. in Mathematics, University of Rhode Island

B.S. in Mechanical Engineering, Cornell University

He uses really creative examples for explaining tough concepts and illustrates them perfectly on the whiteboard. It's impossible to get lost during his lessons.

Thiswas EXCELLENT! I am a math teacher and have been looking for an easy/logical way to explain the lateral area of a cone to my students and this was incredibly helpful, thank you very much!”

I just learned more In 3 minutes of polygons here than I do in 3 weeks in my math class”

Hahaha, his examples are the same problems of my math HW!”

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