$|z_1 z_2| = |z_1| |z_2|$.

Triangle Inequality: $|z_1 + z_2| \le |z_1| + |z_2|$.

Division: $\frac{z_1}{z_2} = \frac{z_1 \bar{z_2}}{|z_2|^2}$.

Polar Form (Euler Formula): $z = |z|(\cos \theta + i \sin \theta) = |z|e^{i\theta}$, where $\theta$ is the argument.

Definition: $a^n = a \cdot a \cdot \dots \cdot a$ ($n$ times).
Rules:
- $a^0 = 1$ (for $a \ne 0$)
- $a^{-n} = \frac{1}{a^n}$
- $a^{1/n} = \sqrt[n]{a}$
- $a^m a^n = a^{m+n}$
- $\frac{a^m}{a^n} = a^{m-n}$
- $(a^m)^n = a^{mn}$
- $(ab)^m = a^m b^m$
- $(\frac{a}{b})^m = \frac{a^m}{b^m}$

Definition: If $N = b^x$, then $\log_b N = x$. (Base $b > 0, b \ne 1$)
Special Cases:
- $\log_b b = 1$
- $\log_b 1 = 0$
- $\log_b (b^n) = n$
- $b^{\log_b a} = a$
Types:
- Natural Logarithm: $\ln x = \log_e x$ (base $e \approx 2.718$)
- Common Logarithm: $\log x = \log_{10} x$ (base $10$)
Properties:
- $\log_b (MN) = \log_b M + \log_b N$
- $\log_b (M^n) = n \log_b M$
- $\log_b (\frac{M}{N}) = \log_b M - \log_b N$
- Change of Base: $\log_{b_2} N = \frac{\log_{b_1} N}{\log_{b_1} b_2}$
Antilogarithm: If $\log_b y = x$, then $y = \text{antilog}_b(x) = b^x$.

Richter Scale for Earthquakes: $R = \log_{10} (\frac{\lambda}{\lambda_0})$
Population Growth/Decay: Exponential models, often involving logarithms to solve for time.
Compound Interest: $A = P(1 + r/n)^{nt}$ (exponential growth)

Set: A well-defined collection of distinct objects.
Elements/Members: Objects belonging to a set. Notation: $a \in A$ (a belongs to A), $b \notin A$ (b does not belong to A).
Representation:
- Roster/Tabular Form: List all elements $\{1, 2, 3\}$.
- Set-Builder Form: Describe properties $\{x : x \text{ is a natural number and } x < 4\}$.
Special Sets:
- $\mathbb{N}$: Natural numbers $\{1, 2, 3, \dots\}$
- $\mathbb{Z}$: Integers $\{\dots, -1, 0, 1, \dots\}$
- $\mathbb{W}$: Whole numbers $\{0, 1, 2, \dots\}$
- $\mathbb{Q}$: Rational numbers
- $\mathbb{R}$: Real numbers

Singleton Set: Contains only one element. E.g., $\{2\}$.
Empty Set ($\emptyset$ or $\{\}$): Contains no elements. Cardinality $n(\emptyset)=0$.
Finite Set: Has a definite (countable) number of elements. $n(A)$ is its cardinality.
Infinite Set: Not finite. Elements cannot be numbered $1$ to $n$.
Equal Sets: $A=B$ if they have exactly the same elements. Order and repetition don't matter.
Subset ($A \subseteq B$): Every element of $A$ is also an element of $B$.
- $\emptyset$ is a subset of every set.
- Every set is a subset of itself.
- Proper Subset ($A \subset B$): $A \subseteq B$ and $A \ne B$.
Power Set ($P(