Regarding definition of exponential function

Question

I would like to know what is the "exact" definition of exponential functions like $f(x)=7^x , 8^x$. I have this doubt when I was thinking about what it means by $7^2$ etc. but what about $7^\pi$ or $7^e$. I have used exponential functions many times. But I have realised that I never knew the "exact" mathematical definition. I know how their curves are made and how to do their functional analysis. But I think I have seen their exact "definition". Can you please quote a source too. I am here because googling gave me useless results.

Answer 1

We define $x^y = \exp(y\log x)$, where $\exp$ is the exponential function and $\log$ is its inverse, the logarithm. It can be shown that the property $$ \exp(a + b) = \exp a \exp b $$ holds for all real $a,b$, and therefore it makes sense to also write $\exp x = e^x$, where we define $e = e^1 = \exp 1$. Let now $n$ be an integer. We have from the property above that $$ x^n = \exp(\log x)^n = \exp(n \log x). $$ The expression on the right hand side makes sense also if $n$ is an arbitrary real number, so this we take as definition of $x^n$ in general.

There are several equivalent definitions of the exponential function. The most popular one is probably $$ \exp x = ∑_{k=0}^∞ \frac{x^k}{k!} = 1 + \lim_{N→∞} ∑_{k=1}^N \frac{x^k}{k!}. $$ This definition contains a limit, so you will typically need real numbers (or complex!) for it to make sense. The logarithm is defined from the inverse function theorem.

Answer 2

I assume your your question is on how to expand $x^n$, whit $n\in \mathbb N$ to $n\in \mathbb R$. Look at $n\in \mathbb Q$:

$x^{n+\frac 1 k}=x^n x^{\frac 1 k}$.

If you understand the way from natural numbers to rational number and from there on, to real numbers, your eyes will open.

Until then, you don't have to understand anything about $\log x$.

After then, understanding of $\log x$ will flow in by itself.