Spin-off from here.
Context: Highschool textbooks often ask students to find the domain of functions. Let's say $f(x) = x+2$. The domain is $\mathbb{R}$...suppose a student (highschool or o/w) asks why $f(x)$ is defined ∀$\mathbb{R}$...?
What about $f(x)=5e^{x}$? $f(x)=\ln(x)$ for $\mathbb{R}^{+}$?
To explain $x+2$ to a high school student (and even as a first pass to a more advanced student), I would appeal our geometric intuition of addition: $x$ is a point on the number line, $x+2$ is the point two units to the right.
To explain $e^x$ (if I may simplify and drop the factor of $5$): first understand $e^n$ for natural numbers using iterated multiplication; then $e^{-n}$ using multiplicative inverses; then $e^{1/n}$ using roots, then $e^{m/n}$ using laws of exponentiation; then plotting some values for various fractions and see how they connect together smoothly to see that $e^x$ can be defined by interpolation.
To explain $\ln(x)$ I would appeal to my now completed explanation of $e^x$, and flip the graph across the line $y=x$.
All of this is done more rigorously in college mathematics, and the explanations are quite a bit more sophisticated. To define arithmetic on the real numbers one could simply appeal to the axioms of the real numbers. But students should also see how those axioms are modelled on the more basic Peano axioms for natural number arithmetic (or at an even more basic level, from the axioms of set theory). Starting from the Peano axioms, one constructs the integers as equivalence classes of pairs of natural numbers $(m,n) \sim (p,q) \iff m+q=n+p$, and one defines their arithmetic and proves the basic laws of commutativity, associativity, identies, inverses, distributive, etc. Then one constructs the rational numbers as equivalence classes of pairs of integers $(a,b) \sim (c,d) \iff ad = bc$, and one defines their arithmetic. Then one constructs the real numbers as equivalence classes of Cauchy sequences of rational numbers (or Dedekind cuts of rational numbers), defines their addition etc.
To define exponentiation and logarithms rigorously, there is nothing better than a good solid calculus course: $\ln(x) = \int_1^x \frac{1}{t} dt$, and the integral exists if and only if $x>0$ so that's the domain. And so on.
