Business Pre-Calculus: Indices, Logarithms, Applications, and Matrix Algebra

Comprehensive practice flashcards covering the laws of indices, logarithmic functions and properties, exponential applications (growth, decay, compound interest), and matrix algebra (matrix types, operations, determinants, inverses, and Cramer's Rule).

What is an index (or exponent) in mathematics?

An index or exponent is a quantity representing the power to which a given base number or expression is raised, written as a raised symbol beside the number (for example, the $3$ in $2^3 = 2 \times 2 \times 2$).

What is the Product Rule for multiplying numbers or expressions in index form with a common base?

$a^m \times a^n = a^{m+n}$, which states that you keep the common base and add the exponents.

What is the Quotient Rule for dividing indices with a common base?

$a^n \div a^m = a^{n-m}$, which states that you keep the common base and subtract the exponent of the divisor from the exponent of the dividend.

What is the Power Rule for raising an index expression to another power?

$(a^m)^n = a^{mn}$, which states that you keep the base and multiply the exponents.

What is the value of any non-zero base raised to the power of zero ($a^0$)?

$a^0 = 1$.

What is the rule for evaluating negative indices ($a^{-m}$)?

$a^{-m} = \frac{1}{a^m}$, meaning any exponent raised to a negative power equals one divided by the base raised to the positive power.

How is a fractional exponent $a^{m/n}$ expressed in radical form?

$a^{m/n} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$.

What is the Property of Equality for Exponential Equations for a common positive base $b$ (where $b \neq 1$)?

Two powers with the same positive base $b$ are equal if and only if their exponents are equal: if $b^x = b^y$, then $x = y$.

What algebraic procedure is used to solve an exponential equation with unlike bases, such as $9^{x+2} = 27^x$?

Rewrite both sides using a common base: $(3^2)^{x+2} = (3^3)^x \rightarrow 3^{2x+4} = 3^{3x}$, then equate the exponents ($2x + 4 = 3x$) and solve to find $x = 4$.

How is an inverse function obtained algebraically and graphically?

Algebraically, an inverse function is the set of ordered pairs obtained by interchanging the $x$ and $y$ values; graphically, it is formed by reflecting the function across the line $y = x$.

What are the domain and range of the basic exponential function $y = a^x$?

Domain: $\{x \in \mathbb{R}\}$; Range: $\{y > 0\}$.

What is the relationship between an exponential function and a logarithmic function?

A logarithmic function is the inverse of an exponential function ($x = a^y \iff y = \log_a(x)$), meaning taking a logarithm undoes exponentiation.

What is the general Change-of-Base Formula for logarithms?

For positive real numbers $a, b, x$ where $a \neq 1$ and $b \neq 1$, $\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$. For common and natural logs, $\log_a(x) = \frac{\log_{10}(x)}{\log_{10}(a)} = \frac{\ln(x)}{\ln(a)}$.

What are the Product, Quotient, and Power Properties of logarithms?

Product Property: $\log_a(uv) = \log_a(u) + \log_a(v)$; Quotient Property: $\log_a\left(\frac{u}{v}\right) = \log_a(u) - \log_a(v)$; Power Property: $\log_a(u^n) = n\log_a(u)$.

What are the three most common real-world applications of exponential functions discussed in Lecture 3?

Population growth, exponential decay, and compound interest.

World population is modeled by $P(t) = 5.94(2^{t/40})$ in billions, where $t$ is years from 1998. What are the predicted populations for 2028 and 1970?

For 2028 ($t = 30$): $10\text{ billion}$; for 1970 ($t = -28$): $3.66\text{ billion}$.

If the number of bacteria in a culture is given by $N(t) = 300\left(\frac{4}{3}\right)^t$, what is the initial count and the count after $3\text{ minutes}$?

Initially ($t = 0$): $300$ bacteria; after $3\text{ minutes}$: approximately $711$ bacteria.

How does the parameter $k$ differ between exponential growth and exponential decay models?

Both models use the same general growth formula structure, but in exponential decay, the value of $k$ is negative (such as in $D = 20e^{-0.4t}$).

What is the compound interest formula and what does each variable represent?

$A = P\left(1 + \frac{r}{n}\right)^{nt}$, where $A$ is final amount, $P$ is principal, $r$ is annual interest rate (as a decimal), $n$ is compoundings per year, and $t$ is time in years.

What are the values of $n$ in the compound interest formula for annual, quarterly, and monthly compounding?

Annually: $n = 1$; Quarterly: $n = 4$; Monthly: $n = 12$.

If growth stocks follow $P(t) = 20(2^{t/10})$ in thousands from 2015, when will the stocks reach $160\text{ thousand}$?

At $t = 30\text{ years}$, which corresponds to the year $2045$.

What is the mathematical definition and order of a matrix?

A matrix is a rectangular array of numbers, symbols, or expressions arranged in rows and columns; its order is given as $m \times n$, where $m$ is the number of rows and $n$ is the number of columns.

What are row and column matrices (vectors)?

A row matrix (or row vector) consists of a single row (e.g., $\begin{pmatrix} 1 & 2 & 3 & 4 \end{pmatrix}$); a column matrix (or column vector) consists of a single column (e.g., $\begin{pmatrix} 1 \\ 3 \\ 2 \end{pmatrix}$).

What is the definition of a square matrix?

A matrix in which the number of rows equals the number of columns ($m = n$).

How are diagonal, scalar, and identity matrices defined and distinguished?

A diagonal matrix is a square matrix whose non-leading diagonal entries are all zero; a scalar matrix is a diagonal matrix where all leading diagonal entries are equal; an identity matrix is a diagonal matrix where all leading diagonal entries are $1$.

What distinguishes an upper triangular matrix from a lower triangular matrix?

In an upper triangular matrix, all entries below the main diagonal are zero; in a lower triangular matrix, all entries above the main diagonal are zero.

What condition defines a symmetric matrix?

A square matrix $A$ is symmetric if $A = A^T$ (or $a_{ij} = a_{ji}$ for all $i, j$), meaning elements across the main diagonal are mirror images of each other.

What two conditions are required for two matrices to be equal?

1) They must have the same dimensions ($m \times n$); 2) Every element in one matrix must be equal to the corresponding element in the other ($a_{ij} = b_{ij}$).

How is the transpose of a matrix $A$ formed, and what are its properties?

The transpose ($A^T$ or $A'$) is formed by swapping rows and columns (row $1$ becomes column $1$). Key properties: $(A')' = A$, $(A + B)' = A' + B'$, and $(kA)' = kA'$.

What condition must be satisfied to multiply two matrices $A$ and $B$?

Matrix multiplication $AB$ is defined only when the number of columns in $A$ equals the number of rows in $B$. If $A$ is of order $m \times k$ and $B$ is of order $k \times n$, the product $AB$ has order $m \times n$.

How is the determinant of a $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$ calculated?

$\det(A) = |A| = ad - bc$.

What is the formula for the inverse of an invertible $2 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \end{pmatrix}$?

$A^{-1} = \frac{1}{ad - bc} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$, provided $\det(A) = ad - bc \neq 0$.

What method for evaluating the determinant of a $3 \times 3$ matrix is depicted in this diagram?

Sarrus' Rule, where the first two columns are rewritten next to the matrix to sum diagonal triple products: $|A| = aqz + brx + cpy - ary - bpz - cqx$.

How is a minor $M_{ij}$ of an entry in a $3 \times 3$ matrix calculated?

By deleting row $i$ and column $j$ of the matrix and evaluating the determinant of the remaining $2 \times 2$ submatrix.

What is the purpose of this alternating sign checkerboard when finding the inverse of a $3 \times 3$ matrix?

It provides the signs to multiply each minor $M_{ij}$ by to produce the cofactor matrix: $C_{ij} = (-1)^{i+j}M_{ij}$.

What is the adjugate (adjoint) matrix, and what is the general formula for the inverse of a $3 \times 3$ matrix?

The adjugate $\text{adj}(A)$ is the transpose of the cofactor matrix ($C^T$); the inverse formula is $A^{-1} = \frac{1}{\det(A)} \text{adj}(A)$, valid only when $\det(A) \neq 0$.

How does Cramer's Rule determine the solution for $x$ in a linear system?

$x = \frac{D_x}{D}$, where $D$ is the determinant of the coefficient matrix and $D_x$ is the determinant of the matrix formed by replacing the $x$-coefficient column with the constants column.