Are the logarithm rules and exponentiation rules (e.g. $a^{x+y}=a^xa^y$) axioms when talking about real numbers?
I know many of them can be proved via induction for integers, but no professor has ever mentioned a proof of these 'rules' when having $\Bbb R$ as a domain.
On
Walter Rudin's Principles of Mathematical Analysis addresses exactly this in Chapter 1. Problem 6. Which outlines:
Fix $b> 1$. If $p = m/n = p/q$ ($m,n,p,q$ integers) prove $(b^m)^{\frac{1}{n}} = (b^p)^{\frac{1}{q}} $.
This shows that $b$ to a rational power is well and consistently defined.
($b^{\frac{1}{n}}$ had been previously defined to be the one postive n-th root which had previously been proven to exist and be unique.)
b) proof $b^{r + s} = b^rb^r $ for rational r, s.
c) And this is the crux.
If x is real, define B(x) to be the set of all numbers $b^t$ where t is rational and t<=x. Prove that
$b^r = sup B(r)$ when r is rational.
This shows we can define $b^x = sup B(x)$ is an extended definition into the reals.
d) Prove $b^{x+y} = b^xb^y$ for real x, y.
Problem 7, is to define the logarithm for real numbers and that it is unique.
Do you want details of the solution?
They can be, but they don't have to be. One of the simplest setups is to define $\exp$ to be the unique function satisfying $\exp'=\exp$ and $\exp(0)=1$; then define $\ln$ to be the inverse of $\exp$; then define $a^x=\exp(x \ln(a))$ for $a>0$. Then the exponent and logarithm rules are theorems. Most of the proof of the rules is just elementary algebra; the tricky part is proving (from the definition above) the functional equation $\exp(x+y)=\exp(x)\exp(y)$.