logarithm

Logarithm

For any positive numbers $M,N$ and positive base $b\ne 1$:
A logarithm tells us what exponent we need to raise the base to get a number:
If $b^y=x$ then ${\log }_b(x)=y$
Written as ${\log }_b(x)$ or $\ln (x)$ for natural log (base $e$)

Basic Properties

${\log }_b(M\cdot N)={\log }_b(M)+{\log }_b(N)$
${\log }_b(\frac{M}{N})={\log }_b(M)-{\log }_b(N)$
${\log }_b(M^n)=n\cdot {\log }_b(M)$
${\log }_b(b)=1$
${\log }_b(1)=0$
${\log }_b(b^n)=n$
${\log }_b(x)=\frac{\ln (x)}{\ln (b)}$ (change of basis)

Logarithm turns multiplication into addition

The logarithm is an isomorphism between $(R^{+},\cdot )$ and $(R,+)$:

$$\log :(R^{+},\cdot )\to (R,+)$$

It transforms multiplication into addition: $\log (xy)=\log (x)+\log (y)$

This satisfies all isomorphism requirements:

• Homomorphism: Preserves group operation (multiplication → addition)
• Bijective: One-to-one correspondence via $\exp$ as inverse
• Identity preservation: $\log (1)=0$
• Inverse preservation: $\log (1/x)=-\log (x)$

Why we want additive instead of multiplicative

Linear algebra works:
Once in log-space, we can use all the tools of linear algebra. Exponential relationships $y=e^b\cdot e^{mx}$ become linear $\log y=mx+b$, so we can use linear regression to find best-fit parameters, generally, matrix operations work on log-transformed data.

Averaging becomes meaningful: Arithmetic mean of log-values $\frac{1}{n}\sum \log x_i=\log (\sqrt[n]{\prod x_i})$ gives the geometric mean - the “typical” multiplicative factor. For growth rates: average $2\times$ and $8\times$ growth isn’t $5\times$ (arithmetic) but $4\times$ (geometric: $\sqrt{2\cdot 8}$).

Domain expansion: Multiplicative operations are trapped in $(0,\infty )$ - can’t represent “negative growth” or pass through zero. Log maps this to $(-\infty ,\infty )$ where we can freely add/subtract without worrying about sign constraints or division by zero.

Numerical stability: Multiplying probabilities like $0.01^{100}$ gives $10^{-200}$ (underflows to 0 in float64). In log-space: $100\cdot \log (0.01)=-200$, perfectly representable. Similarly, $1000000^{100}$ overflows, but $100\cdot \log (10^6)=600$ doesn’t.

Probability & Information:
Independent events have probabilities that multiply: $P(A\cap B)=P(A)\cdot P(B)$
In log-space they add: $\log P(A\cap B)=\log P(A)+\log P(B)$
→ We can sum evidence from multiple sources. Log-likelihood ratios add up. Maximum likelihood becomes a sum to maximize, not a product.
This additivity defines information: $I(x)=-\log P(x)$ bits. Rare events (small $P$) have large positive information.
Entropy $H=-\sum p_i\log p_i$ becomes extensive: entropy of two independent systems equals sum of individual entropies.

Signal processing:
Each amplifier/attenuator multiplies signal: ${\text{signal}}_{\text{out}}=G_1\cdot G_2\cdot G_3\cdot {\text{signal}}_{\text{in}}$
In decibels they add: $20{\log }_{10}(G_1)+20{\log }_{10}(G_2)+20{\log }_{10}(G_3)$ dB
→ Can quickly calculate total system gain by adding component gains. A $1000\times$ amplifier followed by $0.01\times$ attenuator? That’s 60 dB + (-40 dB) = 20 dB = $10\times$ total gain.
Also compresses huge dynamic range: microvolts to kilovolts ($10^9$ range) becomes manageable -180 to +60 dBV.

Chemistry: Hydrogen ion concentration spans $10^{-14}$ to $10^0$ mol/L (100 trillion-fold range!)
pH scale makes this linear: pH 0-14, where each unit = $10\times$ concentration change
Reaction energies add because $\Delta G=-RT\ln K$ converts multiplicative equilibrium constant $K$ to additive free energy

Growth processes: Exponential growth $x(t)=x_0{e}^{rt}$ becomes linear in log: $\log x(t)=\log x_0+rt$
Now it’s a straight line with slope $r$ - can estimate growth rate from any two points
Can use linear regression to fit exponential models, detect deviations from exponential growth