algebra question with exponents

[JeenLeen]

Let's say I have something like e^(a/b), and I want to separate the a from the b. Is there any way to do that?
(If the answer is no, then I'm probably looking at the solution in the wrong way.)

I know e^a * e^b is not the same thing, since that'd be e^(a+b).

Context: we're practicing factorization to find sufficient statistics, and the pdf I'm working with has e^(abs(x)/theta). I want to see if I can separate the Xs from the thetas without doing something like logging everything.

[Knaight]

Sure. Take something like e^(a/b)=c. Take the natural log of both sides and you get a/b=ln(c). Solving for a gets a=b*ln(c).

[Kato]

Not sure if it's the answer your looking for but you can write the denominator in an exponent as its root. So e^(a/b) = b-th root of e^a (sorry, can't type it out better)
Hope this helps.

[georgie_leech]

Not sure if it's the answer your looking for but you can write the denominator in an exponent as its root. So e^(a/b) = b-th root of e^a (sorry, can't type it out better)
Hope this helps.

When not using a math tool, I like superscripts: b√(e^a). And if it's still confusing to readers, an example could be "5^(3/2) means the square root of 5 cubed."

[Caerulea]

So, to follow the above answers through, you might start with:

e^(a/b) = c
Raise both sides to the bth power
e^(ba/b) = c^b
e^a = c^b
And then take the natural log if you actually want to solve for a.

—Caerulea

[JeenLeen]

Another related question, doing this with summations.
if I have something like a summation from i to n of e^x-sub-i (e.g., summing up e^x for a series of X values), I think that's the same as saying e^SUM(x).

If I have SUM(e^(x^2)), would that be e^SUM(x^2) or e^SUM(X)^2, or are those equivalent?

[Khedrac]

Another related question, doing this with summations.
if I have something like a summation from i to n of e^x-sub-i (e.g., summing up e^x for a series of X values), I think that's the same as saying e^SUM(x).

If I have SUM(e^(x^2)), would that be e^SUM(x^2) or e^SUM(X)^2, or are those equivalent?

If you mean what I think you mean then no - you are starting from a false premise.

In general a^b × a^c = a^(b+c)

So, e^x1 + e^x2 + e^x3... DOES NOT = e^(x1+x2+x3...)

and your other sums are not remotely equivalent.

[Lord Torath]

Another related question, doing this with summations.
if I have something like a summation from i to n of e^x-sub-i (e.g., summing up e^x for a series of X values), I think that's the same as saying e^SUM(x).

If I have SUM(e^(x^2)), would that be e^SUM(x^2) or e^SUM(X)^2, or are those equivalent?The quick and easy way to check those is to try plugging some numbers in.

SUM(e^x2) for x= 1,3 = e^12+e^22+e^32 = e1+e4+e9 = 8160.4

e^SUM(x2) = e1+4+9 = e14 = 1,202,604

e^SUM(x)2 = e^(1+2+3)2 = e^62 = e36 = 4.311x1015.

So none of those expressions are equal. SUM(ex) =/= eSUM(x)

Edit: Ninja'd

[JeenLeen]

Thanks, y'all. I think I was misled by some strangely placed or missing parentheses when reading something, then going off in a bad route from there. This clarifies a lot!

EDIT

I had forgotten something, and actually meant to ask about the product instead of the sum.

So... with PRODUCT meaning the product across several X values (e.g., from i=1 to n), would
PRODUCT(e^(X^2)) = e^PRODUCT(X^2) = e^PRODUCT(X)^2

[Lord Torath]

Thanks, y'all. I think I was misled by some strangely placed or missing parentheses when reading something, then going off in a bad route from there. This clarifies a lot!

EDIT

I had forgotten something, and actually meant to ask about the product instead of the sum.

So... with PRODUCT meaning the product across several X values (e.g., from i=1 to n), would
PRODUCT(e^(X^2)) = e^PRODUCT(X^2) = e^PRODUCT(X)^2I'm inclined to say "No" to all three. Again, let's try plugging in some numbers:

Pruduct(e^x2) for x= 1,3 = e^12 x e^22 x e^32 = e1 x e4 x e9 = 1,202,604

e^Product(x2) = e1x4x9 = e36 = 4.311x1015

e^Product(x)2 = e^(1x2x3)2 = e^62 = e36 = 4.311x1015.

Looks like the last two are equivalent (and thinking it through, I can see why: a*a*b*b*c*c = (a*b*c)*(a*b*c)), but the first one is definitely not the same.

[Khedrac]

Thanks, y'all. I think I was misled by some strangely placed or missing parentheses when reading something, then going off in a bad route from there. This clarifies a lot!
You are welcome.

I had forgotten something, and actually meant to ask about the product instead of the sum.

So... with PRODUCT meaning the product across several X values (e.g., from i=1 to n), would
PRODUCT(e^(X^2)) = e^PRODUCT(X^2) = e^PRODUCT(X)^2
Please actually read the posts trying to offer you advice before re-posting a question...


If you mean what I think you mean then no - you are starting from a false premise.

In general a^b × a^c = a^(b+c)
So, looking again at what I posted, what does e^x1 × e^x2 × e^x3 × ... = e^(x1 + x2 + x3 + ...)?

Also, remember a^(b^c) =/= (a^b)^c = a^(b×c)

On the other hand:
e^(x1^2 × x2^2 × x3^2...) = e^((x1 × x2 × x3...)^2) because multiplication is commutative, so your last two terms are identical.

This is getting rather hard to read, so be very careful with your formulae and consider adding in redundant parentheses - they may not be redundant.