How do I calculate the base of an exponent if I know the result and what it was raised to?

Question

If I know the outcome of a value being raised to a certain power, is it possible to know what the original base was?

Example:

x ^ 0.25 = 2.5045

What's the proper way to calculate x?

Answer 1

They way to generally do this would be that you take the logarithm of both sides of the equation. Then you would have:

$$ \log(x ^{0,25} ) = \log(2.5045) $$

Then, since $$ \log(a^b) = b * \log(a) $$

you end up with $$ 0,25 * \log(x) = \log(2.5045) $$

which implies that $$ 10^{(\log(2.5045) / 0,25)} = x $$ by the definition of the logarithm

Answer 2

In this particular example you can rewrite $x^{0.25}=x^{\frac{1}{4}}$

Then you can just put $x=2,5045^{\frac{4}{1}}$

Answer 3

The proper way to do $x^b = a$ is to both sides to $1/b$.

$x = (x^b)^{1/b} = a^{1/b}$.

So $x = (x^{.25})^{1/.25} = 2.5045^{1/.25} = 2.5045^4 = 39.3445102866600625$.