Exponential and Logarithmic Functions¶
Recall (doc/postdoc). Not a first course.
Job: continuous isomorphism \(\exp:(\mathbb{R},+)\xrightarrow{\sim}(\mathbb{R}_{>0},\times)\); inverse \(\ln\); ODE \(y'=y\).
Core¶
\[
\exp_b(x)=b^x,\quad b>0,\,b\neq1;\qquad
\log_b=\exp_b^{-1}:\mathbb{R}_{>0}\to\mathbb{R}
\]
| Property | Formula |
|---|---|
| Functional eq. | \(b^{x+y}=b^xb^y\) |
| Inverse | \(\log_b(b^x)=x\), \(b^{\log_b x}=x\) |
| Change | \(b^x=e^{x\ln b}\), \(\log_b x=\ln x/\ln b\) |
| Derivative | \((e^x)'=e^x\), \((\ln x)'=1/x\) |
| Growth | \(b>1\): incr.; \(0<b<1\): decr. |
Natural base: \(e=\lim(1+1/n)^n=\sum 1/n!\); unique \(a\) with \((a^x)'=a^x\).
Complex: \(e^z\) entire; \(\operatorname{Log}\) multi-valued / branched.
Used later for¶
- Calculus: chain rule models \(e^{kx}\)
- ODEs: \(y'=ky\), linear systems \(e^{tA}\)
- Complex analysis: \(e^{i\theta}\), logs
- Probability: exponential, lognormal, MGF
- Lie groups: one-parameter subgroups
Check¶
Why is there no continuous logarithm \(\mathbb{C}^\times\to\mathbb{C}\) that is a group homomorphism with \(\log(e^z)=z\) globally?
\(\mathbb{C}^\times\) is not simply connected; \(e^z\) is a covering \(\mathbb{C}\to\mathbb{C}^\times\) — no continuous global section through all of \(\mathbb{C}^\times\).