Simple exponential function exercise

[Adyssa]

Homework Statement

is e^x^2 = 4 equivalent to e^x = 2

Homework Equations

As above

The Attempt at a Solution

This is just an exercise, but I'm quite stuck as how to show this is true (or false for that matter). I thought to take the log of both sides and use the log identity to get rid of the double exponent and cancel out the 'ln e' (=1):

ln e^x^2 = ln 4

x^2 ln e = ln 4

x^2 = ln 4

but I'm not sure this helps me (kinda went around in a circle!), nor am I clear on another method to use. I'm sure there's a log property I'm missing? :S

 

[Dick]

e^(x^2) and (e^x)^2 are two different things. e^x^2 doesn't mean much on its own. You probably mean (e^x)^2=4. Try taking ln of that.

 

[Adyssa]

Oh my mistake, I should have been clear, it's e^(x^2) = 4 equiv to e^x = 2.

 

[NascentOxygen]

I think you are asking does e^(x2) = (e^x)²

Let x=some appropriate number, say, 3, and use your calculator.

Re-examining, e^x2 = (e^x)²
you are really asking does x² = 2x

Well, it does if x=2 or 0

 

[Adyssa]

I have to learn how to TEX >.<

The question is to show an equivalence between equation 1 and equation 2 where:

equation 1: e^(x^2) = 4 (so that's e to the x-squared equals 4)

equation 2: e^(x) = 2 (and this is e to the x equals 2)

So I think I have to manipulate equation 1 into the form of equation 2. To be honest, I haven't ever seen a question like this, we didn't do it in class, it's on an (unmarked) exercise quiz relating to basic algebra skills (which I blatantly lack, I keep gettng tripped up on stuff like this, just when I think I understand something!)

I should mention it's a TRUE or FALSE question, so they may not be equivalent, but I'd like to know! :P

 

[NascentOxygen]

If the value of x that solves eqn 2 also solves eqn 1 (using your calculator), then there's a good bet that it is TRUE. Try that for starters.


Your restatement of the eqns amounts to what I wrote, in any case.

 

[NascentOxygen]

I have to learn how to TEX

Make it fun by using with the powerful LaTex graphical generator linked to in the signature at the foot of posts by Jonathon Doolin here: https://www.physicsforums.com/showthread.php?t=525091"

 

[Adyssa]

OK so I did this:

e^x = 2
ln e^x = ln 2 (log law ln e^x = x ln e)
x ln e = ln 2 (ln e = 1, cancels out)
x = ln 2

and sub that into the first equation and sure enough I get 4, so they are equivalent. Now I feel like a right duffer! Thanks for the pointers though, I'm just getting a feel for logs and I get a bit flustered when I don't quite grasp the question.

:)

 

[HallsofIvy]

e^(x^2) and (e^x)^2 are two different things. e^x^2 doesn't mean much on its own. You probably mean (e^x)^2=4. Try taking ln of that.

I disagree. (e^x)^2= e^(2x) which is an easier way to write it. I would immediately assume that e^(x^2) was meant.

And if (e^x)^2= 4 was meant, I would NOT take the logarithm. Since the "outer" function is squaring, I would take the square root first: e^x= +/- 2. Since an exponential (of a real number) cannot be negative, e^x= 2, x= ln(2) is the only (real) solution.

 

[Ray Vickson]

I have to learn how to TEX >.<

The question is to show an equivalence between equation 1 and equation 2 where:

equation 1: e^(x^2) = 4 (so that's e to the x-squared equals 4)

equation 2: e^(x) = 2 (and this is e to the x equals 2)

So I think I have to manipulate equation 1 into the form of equation 2. To be honest, I haven't ever seen a question like this, we didn't do it in class, it's on an (unmarked) exercise quiz relating to basic algebra skills (which I blatantly lack, I keep gettng tripped up on stuff like this, just when I think I understand something!)

I should mention it's a TRUE or FALSE question, so they may not be equivalent, but I'd like to know! :P

No, they are not. The solutions of exp(x^2)=4 are x = +-sqrt(2*ln(2)) = +- 1.1774, while the solution of exp(x)=2 is x = ln(2) = 0.6931. It IS true that x^2 = 2x for some special values of x, but not for those values that solve either of your two equations.

RGV