The rule Symbolab Calculator uses to solve $\sqrt[x]{x^3} = 100$ is not familiar to me and I do not think I have already seen it featuring in any usual exponent/ log rules list
The rule is as follows : $f(x)^{g(x)} = e^{g(x)\times ln(f(x))}$
Is the rule related to the fact that a number , say, $a$, can be expressed as
$e^{ln_e(a)}$ ?
To which ordinary rule can the rule used by Symbolab be reduced? If it can't , how can it be proved or explained?
1 Answer
$\begingroup$Is the rule related to the fact that a number , say, $a$, can be expressed as
$e^{ln_e(a)}$ ?
It is precisely this rule at play, yes. You can use this rule with functions as well:
$$f(x) = e^{\ln(f(x))}$$
Moreover,
$$f(x)^{g(x)} = e^{\ln \left(f(x)^{g(x)} \right)}$$
and, using the property that $\ln(a^b) = b \ln(a)$,
$$f(x)^{g(x)} = e^{g(x) \cdot \ln(f(x))}$$
$\endgroup$
