Exponential and Logarithmic Models: Concepts, Properties, and Applications

Exponential and Logarithmic Models

Exponential Functions

Exponential functions are fundamental in modeling growth and decay processes in calculus. The general form is:

• Standard form: $y = ab^x$ or $y = ae^{kx}$

• a: Initial value

• b: Growth factor

• k: Continuous rate ($k > 0$ for growth, $k < 0$ for decay)

• Relationship: $b = e^k$ and $k = ext{ln}(b)$

Exponential functions are used to model population growth, radioactive decay, and compound interest.

Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are useful for solving equations involving exponents.

• Standard form: $y = ext{log}_b(x)$ or $y = ext{ln}(x)$

• Domain: $x > 0$

• Range: All real numbers

• Vertical Asymptote: $x = 0$

Laws of Logarithms

Properties and Rules

Logarithms follow several key properties that simplify expressions and solve equations:

• Product Rule: $\text{log}_b(MN) = \text{log}_b(M) + \text{log}_b(N)$

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

• Power Rule: $\text{log}_b(M^r) = r \cdot \text{log}_b(M)$

• Change of Base: $\text{log}_b(M) = \frac{\text{log}_a(M)}{\text{log}_a(b)}$

Growth, Decay, and Continuous Compounding

Exponential Growth and Decay

Exponential models describe processes that increase or decrease at rates proportional to their current value.

• General form: $y = ae^{kt}$

• Continuous Interest: $A = Pe^{rt}$

• k > 0: Growth

• k < 0: Decay

• P: Initial amount, r: Rate (decimal), t: Time (years)

Half-Life and Doubling Time

Half-life and doubling time are used to measure the time required for a quantity to halve or double in exponential processes.

• Half-life: $T_{\text{half}} = \frac{\text{ln}(1/2)}{k}$

• Doubling time: $T_{\text{double}} = \frac{\text{ln}(2)}{k}$

Converting Between b and e Models

• $y = ab^x \iff y = ae^{kx}, \quad k = \text{ln}(b)$

Percent Increase / Decrease

• Growth: $b = 1 + r$

• Decay: $b = 1 - r$

Graph Features

Function Type	Domain	Range	Asymptote
Exponential	All real x	$(0, \infty)$	$y = 0$
Logarithmic	$(0, \infty)$	All real y	$x = 0$

Average Rate of Change (AROC)

The average rate of change measures how a function changes over an interval.

• Formula: $\text{AROC} = \frac{f(b) - f(a)}{b - a}$

Worked Examples

Solving Exponential Equations

• Example: Solve $40e^{-0.7t} = 16$

• Divide by 40: $e^{-0.7t} = 0.4$

• Take ln: $-0.7t = \text{ln}(0.4)$

• Solve for t: $t = \frac{\text{ln}(0.4)}{-0.7} \approx 1.31$

Solving Logarithmic Equations

• Example: Solve $\text{log}(5 \cdot 3^x) = 6$

• Convert to exponential form: $5 \cdot 3^x = 10^6$

• $3^x = 2 \times 10^5$

• $x = \frac{\text{log}(2) + 5}{\text{log}(3)} \approx 11.11$

Modeling with $A = Pe^{rt}$

• Example: $4000e^{0.027t} = 9000e^{0.054t}$

• Divide: $e^{0.027t} = 2.25$

• $t = \frac{\text{ln}(2.25)}{0.027} \approx 30$ years

Voltage Decay and Percent Change

• Model: $V(t) = 7e^{-0.16t}$

• (a) $V(6) = 2.68V$

• (b) $V = 1 \rightarrow t = \frac{\text{ln}(1/7)}{-0.16} = 12.16s$

• (c) Percent drop per s $\approx 14.8\%$

Half-Life Problem

• 13% decays in 15 h, 87% remains

• $0.87 = e^{15k} \rightarrow k = \frac{\text{ln}(0.87)}{15} = -0.00932$

• Half-life: $T = \frac{\text{ln}(1/2)}{k} = 74.4$ h

Transcendental Equation

• Example: $\text{ln}(x) = -2x^2$

• Solve numerically or graphically: $x \approx 0.548$

Graphical Intersection

• Example: $e^x = 3x + 2$

• Intersections: $x \approx -0.56$ and $x \approx 2.1$

Log Function Features

• Example: $y = \text{log}(x + 11)$

• Domain: $x > -11$

• Vertical Asymptote: $x = -11$

AROC Example

• Function: $f(x) = 3\text{ln}x$, $1 \leq x \leq 2$

• $\text{AROC} = \frac{3\text{ln}(2) - 3\text{ln}(1)}{2 - 1} = 2.079$

Exponent Rules

Rule	Restriction	Power Rules	Product & Quotient Rules	Zero Exp. Rule	Negative Exp. Rule	Base 1	Neg to Even Power	Neg to Odd Power
$a^0 = 1$	No powers raised to other powers	$(a^m)^n = a^{mn}$	$a^m a^n = a^{m+n}$	$a^0 = 1$	$a^{-n} = \frac{1}{a^n}$	$1^n = 1$	$(-a)^{2n} = a^{2n}$	$(-a)^{2n+1} = -a^{2n+1}$
$a^1 = a$	No parentheses	$(\frac{a}{b})^n = \frac{a^n}{b^n}$	$\frac{a^m}{a^n} = a^{m-n}$

Graphical Features of Exponential and Logarithmic Functions

Function	Shape	Domain	Range	Asymptote
Exponential ($f(x) = 2^x$)	Increasing curve	All real x	$(0, \infty)$	Horizontal: $y = 0$
Logarithmic ($f(x) = \text{log}_2 x$)	Increasing curve	$(0, \infty)$	All real y	Vertical: $x = 0$

For $b > 1$, $y = \text{log}_b x$ increases; for $0 < b < 1$, $y = \text{log}_b x$ decreases.

Properties of Logarithms

Name	Property
Product Rule	$\text{log}_b(mn) = \text{log}_b m + \text{log}_b n$
Quotient Rule	$\text{log}_b\left(\frac{m}{n}\right) = \text{log}_b m - \text{log}_b n$
Power Rule	$\text{log}_b(m^n) = n \cdot \text{log}_b m$

How to Solve Exponential Equations with Logs

1. Isolate the exponential expression(s).

2. If base 10: Take $\text{log}$ of both sides.

3. If NOT base 10: Take $\text{ln}$ of both sides.

4. Use log rules to get $x$ out of the exponent.

5. Solve for $x$.

6. (If asked) Approximate using calculator.

How to Solve Logarithmic Equations: Exponential Form

1. Isolate the log expression.

2. Put in exponential form.

3. Solve for $x$.

4. Check solution by plugging $x$ in $M$.

5. If $M \geq 0$, solution is valid. If $M < 0$, no solution.

Additional info: These notes cover foundational concepts in calculus related to exponential and logarithmic functions, including their properties, graph features, and applications in growth/decay modeling and rate of change. The tables and step-by-step procedures are essential for solving related equations and understanding function behavior.