Sorry for the simple question, but it is so simple that I cannot seem to find the answer!
At school, I was always taught that i can use this identity:
$$a = e^{ln(a)} $$
To rewrite more complex expressions, often in order to get rid of an exponent. For example:
$$\sqrt[4]{(x+1)} = (x+1)^{1/4} = e^{ln[(x+1)^{1/4}]} = e^{\frac{1}{4}ln(x+1)}$$
But nobody ever told me that this can be done if, and only if, the codomain of the original function (the one that we are going to rewrite as an exponential) is always greater than zero. Am I correct in my assumption?
For example, if I were to write:
$$ x^3= e^{3ln(x)}$$
I would be making a mistake, since the last exponential expression $e^{3ln(x)}$ is NOT equal to the first one, $x^3$, because $x^3$ can yield negative values (when $x$ is negative) whereas the codomain of $e^{3ln(x)}$ (or even $e^{3ln(|x|)}$, if we wanted to use the absolute value) is $\mathbb{R^+}$.
Can you please provide any insight to this?
Thanks in advance