QMS110 Lecture 8

Exponential Functions

Definition and Properties

Exponential functions are a fundamental class of functions in business calculus, widely used for modeling growth and decay in real-world scenarios such as finance, population studies, and natural processes.

• Definition: An exponential function is defined as $f(x) = b^x$, where $b > 0$ and $b \neq 1$.

• Domain: The set of all real numbers ($\mathbb{R}$).

• Range: The set of all positive real numbers ($0, \infty$).

• Key Applications:

  • Continuous compounded interest

  • Population growth

  • Radioactive decay

  • Other real-world modeling

Example: For $f(x) = 3^x$, evaluate:

• $f(5) = 3^5 = 243$

• $f\left(-\frac{2}{3}\right) = 3^{-2/3} \approx 0.4807$

• $f(\pi) = 3^{\pi} \approx 31.544$

• $f(\sqrt{2}) = 3^{\sqrt{2}} \approx 4.7288$

Graphing Exponential Functions

Exponential functions can be graphed by plotting points for various values of $x$ and observing the rapid increase or decrease depending on the base.

x	y
-2	0.031
-1	0.125
0	0.50
1	2.00
2	8.00

General Properties:

• If $b > 1$, $f(x) = b^x$ increases rapidly as $x$ increases.

• If $0 < b < 1$, $f(x) = b^x$ decreases rapidly as $x$ increases.

• The $x$-axis ($y=0$) is a horizontal asymptote.

Natural Exponential Function

Base $e$ and Its Importance

The number $e \approx 2.71828$ is an irrational constant and serves as the base for the natural exponential function, which is crucial in calculus and continuous growth models.

• Natural Exponential Function: $f(x) = e^x$

• Domain: $\mathbb{R}$

• Range: $(0, \infty)$

Applications:

• Continuous compound interest

• Population growth and decay

Continuous Compound Interest

When interest is compounded continuously, the formula for the future value $A$ is:

• $A = Pe^{rt}$

• P: Principal (initial amount)

• r: Annual interest rate (decimal)

• t: Time in years

Example: $P = 1,000$, $r = 0.10$, $t = 10$

• $A = 1,000e^{0.10 \times 10} \approx 2,718.18$

Compared to monthly compounding ($A \approx 2,707.04$), continuous compounding yields a slightly higher amount.

Logarithmic Functions

Definition and Properties

Logarithmic functions are the inverses of exponential functions and are essential for solving equations involving exponentials.

• Definition: For $a > 0$, $a \neq 1$, the logarithmic function with base $a$ is $f(x) = \log_a x$.

• Inverse Relationship: $y = \log_a x \iff a^y = x$

• Domain: $(0, \infty)$

• Range: $\mathbb{R}$

• Vertical Asymptote: $x = 0$

Example: $\log_3 81 = x \implies 3^x = 81 \implies x = 4$

Properties of Logarithms

Logarithms have several important properties analogous to exponent rules:

Exponent Law	Logarithm Law
$a^{x+y} = a^x a^y$	$\log_a(xy) = \log_a x + \log_a y$
$a^{x-y} = \frac{a^x}{a^y}$	$\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$
$(a^x)^k = a^{kx}$	$\log_a(x^k) = k \log_a x$

Common and Natural Logarithms

• Common Logarithm: Base 10, denoted $\log x$

• Natural Logarithm: Base $e$, denoted $\ln x$

• Inverse: $y = \ln x$ is the inverse of $y = e^x$

Change of Base Formula

To convert between logarithms of different bases:

• $\log_b a = \frac{\log_k a}{\log_k b}$, where $k$ is any positive base (commonly 10 or $e$)

Example: $\log_5 20 = \frac{\log_{10} 20}{\log_{10} 5} \approx 1.8614$

Expanding and Combining Logarithmic Expressions

• Expansion: $\log_a(xy) = \log_a x + \log_a y$

• Expansion: $\log_a\left(\frac{x}{y}\right) = \log_a x - \log_a y$

• Expansion: $\log_a(x^k) = k \log_a x$

• Combination: $\log_a x + \log_a y = \log_a(xy)$

Example: $2 + \log_2 32 = \log_2(2 \times 32) = \log_2 64 = 6$

Applications: The Law of Forgetting

Logarithmic models can describe phenomena such as memory decay:

• If $P_0$ is the initial performance, after time $t$, $\log P = \log P_0 - c \log(t + 1)$, where $c$ is a constant.

Example: If $P_0 = 90$, $c = 0.2$, after $t = 2$ months: $P = \frac{90}{(2+1)^{0.2}} \approx 72$

Summary Table: Key Properties of Exponential and Logarithmic Functions

Function	Domain	Range	Asymptote
$f(x) = b^x$	$\mathbb{R}$	$(0, \infty)$	$y = 0$
$f(x) = \log_b x$	$(0, \infty)$	$\mathbb{R}$	$x = 0$

Additional info: These notes expand on the original slides by providing full definitions, formulas, and examples for each concept, ensuring a self-contained study guide for Business Calculus students.