Logarithms¶
Recall (doc/postdoc). Not a first course.
Job: inverse of \(b^{(\cdot)}\); change group law from mult to add — later, discrete log hardness and \(\log\) as primitive of \(1/x\).
Core¶
\[
\log_b x=y\iff b^y=x,\qquad b>0,\,b\neq1,\,x>0
\]
| Law | Formula |
|---|---|
| Product | \(\log_b(MN)=\log_b M+\log_b N\) |
| Quotient | \(\log_b(M/N)=\log_b M-\log_b N\) |
| Power | \(\log_b(M^k)=k\log_b M\) |
| Change of base | \(\log_b x=\dfrac{\log_k x}{\log_k b}\) |
| Inverse | \(\log_b(b^x)=x\), \(b^{\log_b x}=x\) |
Standard: \(\ln=\log_e\), \(\log_{10}\), \(\log_2\). Not defined at \(0\) or negatives in \(\mathbb{R}\).
Identity fails: \(\log(M+N)\neq\log M+\log N\).
Used later for¶
- Exp & log functions: continuous \(\exp,\ln\) as \(C^\infty\) inverses
- Calculus: \(\int dx/x=\ln|x|+C\)
- Complex analysis: principal \(\operatorname{Log}\), branch cuts
- Cryptography: discrete log in finite groups
- Information theory: entropy in bits/nats
Check¶
Why is \(\log_b\) not a group homomorphism \((\mathbb{R},+)\to(\mathbb{R},+)\)?
Domain is \((\mathbb{R}_{>0},\times)\); \(\log_b:(\mathbb{R}_{>0},\times)\xrightarrow{\sim}(\mathbb{R},+)\) is the iso. Source is not additive \(\mathbb{R}\).