We can take a base and raise it to an exponent to get a result, e.g. $2^5 = 32$. This is basically from a power or exponential function, which fixes the exponent or base and takes as input the other, and outputs some result. We can also take logarithms, e.g. $\log_2 32 = 5$, which basically fixes a base and takes as input a number (i.e. the result) and outputs the exponent.
My question is, is there a name for a function that takes the exponent and the result to give the base? It seems that I have not really encountered this, or maybe I just don’t have a name for it. If this is not as common, is there a rough explanation for why it is less common than, say, the power, exponential, or logarithm functions?
Best Answer
For the general equation $$a^b=c$$ (where everything is defined, so no $0^0$ for example, and if you are dealing with real numbers only there are more restrictions), raising both sides to the power of $\frac1b$ will give $a$ (the base) in terms of $b$ and $c$ (the exponent and the result respectively).
$$ \begin{align*} a^b&=c\\ \left(a^b\right)^{(1/b)}&=c^{(1/b)}\;\;\;\text{rasing both sides to the power of $\frac1b$}\\ a^1&=c^{(1/b)}\;\;\;\text{multiplying the powers $b\times\frac1b=1$}\\ a&=c^{(1/b)} \end{align*} $$ Raising to the bower of $\frac1b$ is equivalent to taking the $b$th root, so the above solution can also be written as $$a=\sqrt[b]{c}$$ So the name of the function is the root, specifically the $b$th root in this case.