Exponential and Logarithmic Functions in Business Calculus

Exponential and Logarithmic Functions

Exponential Functions and Compound Interest

Exponential functions are fundamental in modeling growth and decay processes, especially in financial contexts such as compound interest. The general form of an exponential function is:

• Definition: An exponential function has the form $f(x) = a^{x}$, where $a > 0$ and $a \neq 1$.

• Compound Interest Formula: If a principal $P$ is invested at an interest rate $r$ compounded $n$ times per year, after $t$ years the amount $A$ is:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

• Special Cases:

  • Compounded Annually: $A = P(1 + r)^t$

  • Compounded Quarterly: $A = P\left(1 + \frac{r}{4}\right)^{4t}$

  • Compounded Monthly: $A = P\left(1 + \frac{r}{12}\right)^{12t}$

  • Compounded Daily: $A = P\left(1 + \frac{r}{365}\right)^{365t}$

• Continuous Compounding: When compounding occurs infinitely often, the formula becomes:

$A = Pe^{rt}$

• Example: John invests $5000 at 3.25% annual interest compounded continuously for 5 years. The future value is:

$A = 5000 \times e^{0.0325 \times 5}$

• Application: Continuous compounding is used in advanced financial calculations and models.

The Number e and Natural Logarithms

The mathematical constant e (approximately 2.71828) is the base of natural logarithms and is crucial in continuous growth models.

• Definition: $e = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n$

• Natural Logarithm: The natural logarithm, denoted $\ln(x)$, is the inverse function of $e^x$.

• Properties:

  • $\ln(e) = 1$

  • $\ln(1) = 0$

  • $\ln(ab) = \ln(a) + \ln(b)$

  • $\ln\left(\frac{a}{b}\right) = \ln(a) - \ln(b)$

  • $\ln(a^k) = k \ln(a)$

• Example: $\ln(2e^5) = \ln(2) + \ln(e^5) = \ln(2) + 5$

Solving Exponential and Logarithmic Equations

Solving equations involving exponentials and logarithms is essential for applications such as determining investment growth or decay rates.

• Exponential Equations: To solve $e^{x} = b$, take the natural logarithm of both sides:

$x = \ln(b)$

• Logarithmic Equations: To solve $\ln(x) = a$, exponentiate both sides:

$x = e^{a}$

• Example: Marisa invests $12,000 at 2.8% annual interest compounded continuously. To find when her account doubles:

Set $A = 2P$ and use $A = Pe^{rt}$:

$2P = P e^{0.028t}$

$2 = e^{0.028t}$

$\ln(2) = 0.028t$

$t = \frac{\ln(2)}{0.028}$

• Application: This method is used to determine the time required for investments to reach a target value.

Summary Table: Compound Interest Formulas

The following table summarizes the main compound interest formulas for different compounding frequencies:

Compounding Frequency	Formula
Annually	$A = P(1 + r)^t$
Quarterly	$A = P\left(1 + \frac{r}{4}\right)^{4t}$
Monthly	$A = P\left(1 + \frac{r}{12}\right)^{12t}$
Daily	$A = P\left(1 + \frac{r}{365}\right)^{365t}$
Continuously	$A = Pe^{rt}$

Additional info:

• Some formulas and examples were inferred and expanded for clarity and completeness.

• Properties of logarithms and exponential equations were organized for academic context.