# Solving for a variable that's an exponent

How would you figure out this?

$x^y = z$

How do you find out $y$ if you know $x$ and $z$ ?

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Clealry, not... en.wikipedia.org/wiki/Logarithm – Alon Amit May 4 '11 at 21:16

This section on wikipedia explains how.

You take the logarithm of both sides, \begin{align*} x^y=z &\implies \log{x^y}=\log{z}\\ &\implies y\log{x}=\log{z}\\ &\implies y=\frac{\log{z}}{\log{x}}=\log_x{z}. \end{align*} The second and third implications follow by standard rules for logarithms.

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X^Y=Z by definition is LogX^Z = Y (sorry, don't; know the latex codes) – user697111 May 5 '11 at 2:55

## Integers

Divide $z$ by $x$ until you get 1, how many times did it take?: $y$ times.

I'll give an example, $343 = 7^{something}$, but what value is "something"?

1. $\frac{343}{7} = 49$,

2. $\frac{49}7 = 7$,

3. $\frac77 = 1$

...so it's 3. Three separate divisions by 7 lead to one, so 7 x 7 x 7 = 343.

therefore $7^3 = 343$.

## Polynomials

If you suspect $q = p^k$ but don't know $k$ or $q$ you can easily find it out by taking the greatest common divisor of the derivative of $q$ with $q$.

## Reals and complex numbers

The logarithm function is defined by this equation

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