If y=e^x then what does x equal? (Getting 2 diff. answers!)

[Goistein]

I was trying to derive the derivative of e^x, did some random stuff, and arrived to this:

y=e^x

ln both sides

ln(y)=x

Differentiate

dy/y=dx

Multiply by y

dy=ydx

Integrate (When integrating y with respect to x, it becomes a constant)

y=xy

Divide by y

1=x

Well? Where was the mistake?

 

[tkhunny]

Re: If y=e^x then what does x equal?

Very bad.

ln both sides

What does that even mean?

dy=ydx

Integrate (When integrating y with respect to x, it becomes a constant)

y=xy

You state up front that \(\displaystyle y = e^{x}\). More generally, y = f(x). After stating this, you cannot later just play like 'y' is NOT a function of 'x'. It is NOT a constant, here.

You are overlooking a constant of integration. Even if you were right about 'y' being constant, you should be left with y = xy + C ==> 1 = x + C/y. Quite a different animal.

 

[stapel]

dy=ydx

Integrate (When integrating y with respect to x, it becomes a constant)

You might want to study the topic of "separation of variables". Your idea is interesting, but I'm afraid it's simply not how the process works. Sorry!

Eliz.

 

[Goistein]

Re: If y=e^x then what does x equal?

Guess I goofed on this one.

tkhunny: ln means natural logarithm.

 

[tkhunny]

Re: If y=e^x then what does x equal?

I wasn't questioning the use of 'ln' to mean "Natural Logarithm".

I was questioning the the operation "ln both sides". That is absolutely senseless. State it long-hand. "Natural Logarithm both sides." What? "ln" isn't a verb.

 

[Goistein]

This is not an english help site, so we can use bad grammar right? If you were talking to someone, would you say ln both sides, or take the natural logarithm of both sides? I see it as just a shorter way.

 

[stapel]

If you were talking to someone, would you say ln both sides, or take the natural logarithm of both sides?

Since "ell-enn both sides" could be confusing (what the heck does it mean?), and since legitimate tutors would like to see students confident in their knowledge and understanding, I personally would prefer to use the clear terminology. Why invent my own lingo and require everybody else to read my mind to try to figure out what I mean? Why befuddle a student with what sounds like a new thing to learn?

It hardly seems polite. And requiring everybody to ask, repeatedly, what on earth I mean, hardly seems the "quick" way to do things. :shock:

Just as one squares both sides (not "high two's it"), adds a value to each side (not "fours 'em"), and raises both sides to a fractional power (not "do the number up here thingy"), so also one takes the log of both sides.

This is not an english help site, so we can use bad grammar right?

You're welcome to speak as confusingly as you like! :wink:

But don't complain when other people don't know what you mean!

Eliz.