Exponential and Logarithmic Functions: Precalculus Study Notes (MTH 161, Chapter 4)

Exponential Functions

Definition and Properties

An exponential function is a function of the form $f(x) = a^x$, where $a$ is a positive constant and $x$ is a variable exponent. Exponential functions model situations where growth or decay occurs at a constant percentage rate over time.

• Constant base, variable exponent: The base $a$ remains fixed, while the exponent $x$ changes.

• Rapid growth: Exponential growth quickly outpaces polynomial growth as $x$ increases.

• Domain: $(- , )$

• Range: $(0, )$

• Y-intercept: $(0, 1)$

• No x-intercept

• Horizontal asymptote: $y = 0$

• Increasing/Decreasing: If $a > 1$, $f(x)$ increases; if $0 < a < 1$, $f(x)$ decreases.

• One-to-one: Exponential functions are one-to-one and have inverses.

Example: Doubling Deposits

Suppose you deposit $0.01$ on day 1 and double the deposit each day. The amount on day $n$ is $A(n) = 0.01 imes 2^{n-1}$.

• On day 30: $A(30) = 0.01 imes 2^{29} ightarrow$ a very large amount due to exponential growth.

Graphing Exponential Functions

For $f(x) = 2^x$:

x	$2^x$
-2	0.25
-1	0.5
0	1
1	2
2	4
3	8

The graph rises steeply for positive $x$ and approaches zero for negative $x$.

Exponential Growth vs. Polynomial Growth

Exponential functions like $g(x) = 2^x$ grow much faster than polynomial functions such as $k(x) = x^2$.

Applications: Exponential Decay

Exponential decay models processes where quantities decrease at a constant percentage rate, such as radioactive decay.

• General formula: $Q(t) = Q_0 a^t$

• Example: Carbon-14 decay, with half-life $t_{1/2} = 5715$ years: $Q(t) = 10 imes (1/2)^{t/5715}$

Simple and Compound Interest

Simple Interest

Simple interest is calculated only on the principal amount.

• Formula: $I = Prt$

• Future value: $A = P + Prt = P(1 + rt)$

Example

If $P = 500$, $r = 0.12$, $t = 2$:

• $A = 500(1 + 0.12 imes 2) = 620$

Compound Interest

Compound interest is calculated on both the principal and accumulated interest.

• Formula: $A = P igg(1 + rac{r}{n}igg)^{nt}$

• $n$ = number of compounding periods per year

Example

$P = 1000$, $r = 0.045$, $n = 4$, $t = 5$:

• $A = 1000 igg(1 + rac{0.045}{4}igg)^{4 imes 5} = 1250.75$

Continuous Compounding

Interest compounded infinitely often is called continuous compounding.

• Formula: $A = Pe^{rt}$

• $e 2.71828$ (Euler's number)

Example

$P = 1000$, $r = 0.045$, $t = 5$:

• $A = 1000e^{0.045 imes 5} = 1225.32$

Exponential Growth and Decay Models

General Model

For population or radioactive decay:

• $f(t) = a_0 e^{kt}$, where $k$ is the relative rate of growth ($k > 0$) or decay ($k < 0$).

Example

If a population of 8000 grows at 10% per year, after 10 years:

• $A = 8000e^{0.10 imes 10} 22000$

Logarithmic Functions

Definition and Properties

A logarithmic function is the inverse of an exponential function. For $a > 0$, $a 1$, the logarithmic function with base $a$ is $f(x) = ext{log}_a x$.

• Domain: $(0, )$

• Range: $(- , )$

• x-intercept: $(1, 0)$

• No y-intercept

• Vertical asymptote: $x = 0$

• Increasing/Decreasing: If $a > 1$, $f(x)$ increases; if $0 < a < 1$, $f(x)$ decreases.

• One-to-one: Logarithmic functions are one-to-one and have inverses.

Exponential and Logarithmic Functions as Inverses

• If $f(x) = a^x$, then $f^{-1}(x) = ext{log}_a x$

Logarithmic Equations and Properties

• Change of base: $ext{log}_a x = rac{ ext{log}_b x}{ ext{log}_b a}$

• Product property: $ext{log}_a (xy) = ext{log}_a x + ext{log}_a y$

• Quotient property: $ext{log}_a igg( rac{x}{y}igg) = ext{log}_a x - ext{log}_a y$

• Power property: $ext{log}_a (x^r) = r ext{log}_a x$

Examples

• $ext{log}_2 8 = 3$ because $2^3 = 8$

• $ext{log}_2 0.5 = -1$ because $2^{-1} = 0.5$

Graphing Logarithmic Functions

x	$ext{log}_2 x$
0.25	-2
0.5	-1
1	0
2	1
4	2
8	3

Exponential and Logarithmic Functions: Comparison

Exponential ($y = a^x$)	Logarithmic ($y = ext{log}_a x$)
Domain: $(- , )$	Domain: $(0, )$
Range: $(0, )$	Range: $(- , )$
Y-intercept: $(0, 1)$	X-intercept: $(1, 0)$
Horizontal asymptote: $y = 0$	Vertical asymptote: $x = 0$
Increasing if $a > 1$	Increasing if $a > 1$
Decreasing if $0 < a < 1$	Decreasing if $0 < a < 1$
One-to-one, has inverse	One-to-one, has inverse

Solving Exponential and Logarithmic Equations

Exponential Equations

• To solve $a^x = b$, take logarithms of both sides: $x = ext{log}_a b$

Logarithmic Equations

• To solve $ext{log}_a x = y$, rewrite as $x = a^y$

Example

• Find $x$ if $2^x = 8$: $x = ext{log}_2 8 = 3$

• Find $x$ if $ext{log}_2 x = 4$: $x = 2^4 = 16$

Applications and Modeling

Population Growth

• Exponential models predict rapid increases in population or investments over time.

• Linear models predict steady, constant increases.

Radioactive Decay

• Exponential decay models the decrease in radioactive isotopes over time.

Interest Calculations

• Simple interest grows linearly; compound interest grows exponentially.

• Continuous compounding uses the natural exponential function $e^x$.

Summary Table: Key Formulas

Concept	Formula
Exponential Function	$f(x) = a^x$
Logarithmic Function	$f(x) = ext{log}_a x$
Simple Interest	$A = P(1 + rt)$
Compound Interest	$A = P igg(1 + rac{r}{n}igg)^{nt}$
Continuous Compounding	$A = Pe^{rt}$
Exponential Growth/Decay	$f(t) = a_0 e^{kt}$

Additional info: These notes are based on "Precalculus: A Right Triangle Approach, 4th ed." by Ratti, McWaters, and Skrzpek, and cover the essential concepts of exponential and logarithmic functions, their properties, graphs, and applications in finance and science.