Not homework - trying to figure something out for work. How do I use a scientific calculator to figure out what power I have to raise a number by to get another number? Illustration: say I know that the answer is 256 and I want to know what power of 2 gets me there. Yes, I can work it out to being 2^8, but how do I use a scientific calculator to figure that out for something that isn’t as obvious.
I have a feeling I’m going to do a big :smack: once I know the answer, but there it is! 
You just discovered the concept of logarithms.
The log function is what you’re looking for, if your calculator has one.
If x[sup]y[/sup] = z, then (log z)/(log x) = y is probably what you’re looking for.
Ah yes! It’s only been about 25 years since I thought about logs! Thank you very much. I was trying to use the x root of y key.
Just knowing that it’s a log is the first step, but in practice you also need the formula Asymptotically fat gave, since most calculators only have a button for the log base e (generally labeled ln) and base 10 (generally just labeled log). For the log in any other base, you need to use log(z)/log(x) (and it doesn’t matter what base you use for those logs, as long as it’s the same).
In the interest of completeness, I should add that (ln z)/(ln x) = y would also work.
The derivation, in case you care:
x[sup]y[/sup] = z
log (or ln) x[sup]y[/sup] = log (or ln) z
y log x = log z
y = (log z)/(log x)
Not all that useful a derivation, though, because it relies on you knowing that log(x[sup]y[/sup]) = y*log(x). It’s just replacing one mysterious log identity with a different mysterious log identity.
No, that is the definition of logarithms, not a mysterious log identity.
The definition of logarithms is “The power to which a base, such as 10, must be raised to produce a given number.”
your calc might also have a log[sub][/sub] thing which will save you having to type that formula in
Logarithms are therefore indices, from which we deduce from known laws of indices that log x[sup]y[/sup] = y log x; with numbers expressed in index form, you multiply by adding indices, you raise to powers by multiplying indices.
Does the word “index”, in this context, have any meaning beyond being a synonym for “logarithm”?
Well, that’s one way to define logarithms. Of course, different people may phrase their definitions differently, or even use wildly disparate (but, in suitable common contexts, equivalent) definitions. It wouldn’t be wrong to take the property “log(x[sup]y[/sup]) = y log(x)” as one’s definition of a logarithmic function, if one wanted to.
But, anyway, for an audience for whom rules are more familiar when phrased in terms of exponentiation than in terms of logarithms, we might make the observation as follows: recall that (A[sup]B[/sup])[sup]C[/sup] = A[sup]B * C[/sup]. [If you have C many groups of B many groups of factors of A, it’s as good as having B * C many groups of factors of A]
Thus, the power to which you must raise A to get a particular value is B times the power to which you must raise A[sup]B[/sup] to get that same value. That is, log[sub]A/sub = log[sub]A[sup]B[/sup]/sub * B.
So logarithms in any two different bases are the same, except for a multiplicative factor. And what’s the factor? It’s the power to which you have to raise the one base to get the other (i.e., another logarithm).
Thus, we get our change of base formula: If we abbreviate A[sup]B[/sup] as D, so that B = log[sub]A/sub, our last identity becomes log[sub]A/sub = log[sub]D/sub * log[sub]A/sub, which is equivalent to log[sub]A/sub/log[sub]A/sub = log[sub]D/sub.
[None of this is really any different than our starting observation that (A[sup]B[/sup])[sup]C[/sup] = A[sup]B * C[/sup]; we’ve just rephrased it in terms of "To which power do I have to raise … to to get …?"s instead of "What do I get when I raise … to the …th power?"s now, just in case you happen to be talking in that manner already.]
That should work just as well as logarithms, and with far fewer keystrokes. if N = xth root of y, then y to the N would give you x.
It would use fewer keystrokes, but it would generally give you the wrong answer.
Not for the kind of thing the OP is asking about. For example, you can’t use the “xth root of y” key to solve 4[sup]x[/sup] = 8192. Neither of the given numbers tells you what kind of root the xth root is.
Technically no, but many people would use whole-number indices and the rules for multiplication, division and raising to powers without either using the word “logarithm” or any awareness that there could be fractional indices or any possible value of x that would allow 10[sup]x[/sup] = 3, for instance.
An under-14s (year 9, 8th grade, take your pick) class I was with the other day was happily tackling standard-form representation and the arithmetic of same without logarithms ever getting a mention.