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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$\begingroup$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.
$\endgroup$ 1 $\begingroup$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"?
$\frac{343}{7} = 49$,
$\frac{49}7 = 7$,
$\frac77 = 1$
...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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