Applications of Exponential and Logarithmic Functions

Section 5.5 Applications of Exponential and Logarithmic Functions

Review of Converting Between Decimal and Percent Notation

Understanding how to convert between decimals and percents is essential for interpreting exponential and logarithmic models in real-world contexts.

• Decimal to Percent: Multiply the decimal by 100. This shifts the decimal point two places to the right. Example: $0.25 \times 100 = 25\%$

• Percent to Decimal: Divide the percent by 100. This shifts the decimal point two places to the left. Example: $75\% \div 100 = 0.75$

Review of Approximating Exponential Expressions with the Calculator

Calculators can be used to evaluate exponential expressions efficiently. Refer to your calculator's manual for specific instructions.

Review of Solving Exponential Equations of the Form $b^x = c$

Solving exponential equations is a key skill in precalculus, especially when modeling growth and decay.

• To solve $b^x = c$, where $b > 0$ and $b \neq 1$, and $c$ is a constant not equal to zero:

• Rewrite the equation in logarithmic form using the definition of a logarithm: $x = \log_b c$

• Alternatively, use the property of equality to "take the log of both sides" (base 10 or base $e$): $\log(b^x) = \log(c)$, so $x \log(b) = \log(c)$, and $x = \frac{\log(c)}{\log(b)}$

• Solving for $x$ often requires using logarithms to "bring down" any exponents.

Objective 1: Solving Compound Interest Applications

Periodic Compound Interest Formula

Compound interest is a common application of exponential functions in finance.

• The formula for periodic compounding is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ where:

  • $A$ = total amount after $t$ years

  • $P$ = principal (initial investment)

  • $r$ = annual interest rate (as a decimal)

  • $n$ = number of times interest is compounded per year

  • $t$ = number of years

• Example: If $P = 1000$, $r = 0.05$, $n = 4$, $t = 3$, then $A = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 3}$

Continuous Compound Interest Formula

• For continuous compounding, use: $A = Pe^{rt}$ where $e$ is Euler's number ($\approx 2.718$).

• Example: If $P = 500$, $r = 0.06$, $t = 5$, then $A = 500e^{0.06 \times 5}$

Objective 2: Exponential Growth and Decay Applications

Uninhibited Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value, without any limiting factors.

• The model is: $P(t) = P_0 e^{rt}$ where:

  • $P(t)$ = population at time $t$

  • $P_0$ = initial population

  • $r$ = relative growth rate (can be given as a percent or decimal)

• Example: If $P_0 = 100$, $r = 0.03$, $t = 10$, then $P(10) = 100e^{0.3}$

Uninhibited Exponential Decay

Exponential decay models describe quantities that decrease at a rate proportional to their current value, such as radioactive substances or depreciation.

• The model is: $A(t) = A_0 e^{-kt}$ where:

  • $A(t)$ = amount at time $t$

  • $A_0$ = initial amount

  • $k$ = relative decay constant (can be given as a percent or decimal)

• Example: If $A_0 = 200$, $k = 0.05$, $t = 8$, then $A(8) = 200e^{-0.4}$

Half-Life

The half-life of a substance is the time required for half of the initial amount to decay.

• If $A_0$ is the initial amount, then after one half-life, $A = 0.5A_0$

• The half-life formula is: $A = A_0 (0.5)^{t/h}$ where $h$ is the half-life period.

• Example: If $A_0 = 80$ grams, $h = 5$ years, after $t = 10$ years: $A = 80 (0.5)^{10/5} = 80 (0.5)^2 = 80 \times 0.25 = 20$ grams

Summary Table: Exponential Models

Model	Formula	Key Parameters	Application
Periodic Compound Interest	$A = P \left(1 + \frac{r}{n}\right)^{nt}$	$P$, $r$, $n$, $t$	Bank accounts, loans
Continuous Compound Interest	$A = Pe^{rt}$	$P$, $r$, $t$	Finance, investments
Exponential Growth	$P(t) = P_0 e^{rt}$	$P_0$, $r$, $t$	Population, bacteria
Exponential Decay	$A(t) = A_0 e^{-kt}$	$A_0$, $k$, $t$	Radioactive decay, depreciation
Half-Life	$A = A_0 (0.5)^{t/h}$	$A_0$, $h$, $t$	Radioactive substances