Unit 2 Exponents and Logarithms - Exponential Functions

A set of key vocabulary terms, exponent rules, exponential graph characteristics, and exponential problem-solving formula definitions based on Unit 2 Lesson 1.

Exponential Function

A function that has a variable as part of an exponent, such as $f(x) = 2^x$ or $f(x) = 5(3)^x + 4$.

Product Law of Exponents

An exponent law stating that $(x^m)(x^n) = x^{m+n}$.

Quotient Law of Exponents

An exponent law stating that $x^m \times x^n = x^{m-n}$.

Power of a Power Law

An exponent law stating that $(x^m)^n = x^{mn}$.

Power of a Product Law

An exponent law stating that $(xy)^n = x^n y^n$.

Negative Exponent Law

An exponent law stating that $x^{-m} = \frac{1}{x^m}$, $\frac{1}{x^{-m}} = x^m$, and $\left(\frac{x}{y}\right)^{-m} = \left(\frac{y}{x}\right)^m$.

Rational Exponent Law

An exponent law stating that $x^{\frac{1}{n}} = \sqrt[n]{x}$ and $x^{\frac{m}{n}} = \sqrt[n]{x^m} = (\sqrt[n]{x})^m$.

Zero Exponent Law

An exponent law stating that $x^0 = 1$.

$y$-intercept of $y = a^x$

The point $(0, 1)$ where the graph of $y = a^x$ ($a > 0, a \neq 1$) intersects the vertical axis.

Horizontal Asymptote of $y = a^x$

A line located at $y = 0$ (the $x$-axis) that the graph of $y = a^x$ approaches.

Exponential Growth

An exponential relation where $a > 1$ in $y = a^x$, or where the multiplication factor $m > 1$ in $A = P(m)^{\frac{t}{T}}$.

Exponential Decay

An exponential relation where $0 < a < 1$ in $y = a^x$, or where the multiplication factor $0 < m < 1$ in $A = P(m)^{\frac{t}{T}}$.

Variable $A$ (Exponential Formula)

Represents the amount after a specified time in the general formula $A = P(m)^{\frac{t}{T}}$.

Variable $P$ (Exponential Formula)

Represents the principal or initial amount in the general formula $A = P(m)^{\frac{t}{T}}$.

Variable $m$ (Exponential Formula)

Represents the multiplication factor in the general formula $A = P(m)^{\frac{t}{T}}$.

Variable $t$ (Exponential Formula)

Represents the time expired in the general formula $A = P(m)^{\frac{t}{T}}$.

Variable $T$ (Exponential Formula)

Represents the half-life time, doubling period, or tripling period in the general formula $A = P(m)^{\frac{t}{T}}$, which must be in the same units as time expired $t$.