Precalculus Study Guide: Exponential, Logarithmic, Piecewise, Conic Sections, Matrices, Sequences, Induction, and Binomial Theorem

Exponential and Logarithmic Functions

Properties of Logarithms

Logarithms are mathematical functions that are the inverse of exponentiation. Understanding their properties is essential for solving equations and simplifying expressions.

• Product Rule: $\log_b(MN) = \log_b(M) + \log_b(N)$

• Quotient Rule: $\log_b\left(\frac{M}{N}\right) = \log_b(M) - \log_b(N)$

• Power Rule: $\log_b(M^k) = k \cdot \log_b(M)$

• Change of Base Formula: $\log_b(M) = \frac{\log_a(M)}{\log_a(b)}$

Example: Simplify $\log_2(8) + \log_2(4)$ using the product rule.

Since $8 \times 4 = 32$, $\log_2(8) + \log_2(4) = \log_2(32) = 5$.

Exponential and Logarithmic Equations

Solving exponential and logarithmic equations often involves applying logarithmic properties or exponent rules to isolate the variable.

• Exponential Equation: $a^x = b$ → $x = \log_a(b)$

• Logarithmic Equation: $\log_b(x) = y$ → $x = b^y$

Example: Solve $2^x = 16$. Since $16 = 2^4$, $x = 4$.

Applications of Exponential and Logarithmic Functions

These functions are used in modeling growth and decay, such as population growth, radioactive decay, and compound interest.

• Exponential Growth: $y = a e^{kt}$

• Exponential Decay: $y = a e^{-kt}$

Example: The population of bacteria doubles every hour. If there are 100 bacteria initially, after 3 hours: $100 \times 2^3 = 800$ bacteria.

Piecewise Functions

Definition and Examples

Piecewise functions are defined by different expressions depending on the input value. They are useful for modeling situations where a rule changes based on conditions.

• Definition: $f(x) = \begin{cases} x^2 & x < 0 \\ 2x + 1 & x \geq 0 \end{cases}$

Example: For $x = -2$, $f(-2) = (-2)^2 = 4$. For $x = 3$, $f(3) = 2 \times 3 + 1 = 7$.

Conic Sections

Circle, Ellipse, Parabola, Hyperbola

Conic sections are curves obtained by intersecting a plane with a cone. The main types are circles, ellipses, parabolas, and hyperbolas.

• Circle: $x^2 + y^2 = r^2$

• Ellipse: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$

• Parabola: $y = ax^2 + bx + c$

• Hyperbola: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

Example: The equation $x^2 + y^2 = 9$ represents a circle with radius 3.

Matrices and Determinants

Basic Operations and Applications

Matrices are rectangular arrays of numbers used to solve systems of equations and perform linear transformations. Determinants are scalar values that can be computed from square matrices and are used to determine invertibility.

• Addition: Add corresponding elements.

• Multiplication: Multiply rows by columns.

• Determinant of 2x2: $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$, $\text{det} = ad - bc$

Example: $\begin{pmatrix} 1 & 2 \\ 3 & 4 \end{pmatrix}$, $\text{det} = 1 \times 4 - 2 \times 3 = 4 - 6 = -2$.

Sequences, Series, Mathematical Induction, and Binomial Theorem

Sequences and Series

A sequence is an ordered list of numbers, and a series is the sum of a sequence. Common types include arithmetic and geometric sequences.

• Arithmetic Sequence: $a_n = a_1 + (n-1)d$

• Geometric Sequence: $a_n = a_1 r^{n-1}$

• Sum of Arithmetic Series: $S_n = \frac{n}{2}(a_1 + a_n)$

• Sum of Geometric Series: $S_n = a_1 \frac{1 - r^n}{1 - r}$, $r \neq 1$

Mathematical Induction

Mathematical induction is a proof technique used to establish the truth of an infinite sequence of statements.

• Base Case: Prove the statement for $n = 1$.

• Inductive Step: Assume true for $n = k$, prove for $n = k+1$.

Binomial Theorem

The binomial theorem provides a formula for expanding powers of binomials.

• Formula: $(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$

Example: Expand $(x + y)^3$ using the binomial theorem: $x^3 + 3x^2y + 3xy^2 + y^3$.

Summary Table: Main Precalculus Topics Covered

Topic	Key Concepts	Example Formula
Exponential & Logarithmic Functions	Properties, Equations, Applications	$y = a e^{kt}$
Piecewise Functions	Definition, Evaluation	$f(x) = \begin{cases} x^2 & x < 0 \\ 2x + 1 & x \geq 0 \end{cases}$
Conic Sections	Circle, Ellipse, Parabola, Hyperbola	$x^2 + y^2 = r^2$
Matrices & Determinants	Operations, Determinant	$\text{det} = ad - bc$
Sequences & Series	Arithmetic, Geometric, Sums	$a_n = a_1 + (n-1)d$
Mathematical Induction	Proof Technique	Base & Inductive Step
Binomial Theorem	Expansion of Binomials	$(x + y)^n = \sum_{k=0}^{n} \binom{n}{k} x^{n-k} y^k$