C6

Exponential and Logarithmic Functions

Domain, Asymptotes, and Graph Transformations

Exponential and logarithmic functions are fundamental in modeling growth and decay processes. Understanding their domains, asymptotes, and transformations is essential for graphing and interpreting these functions.

• Domain of Logarithmic Functions: The domain of $f(x) = \log_b(x - h)$ is $x > h$.

• Vertical Asymptote: For $f(x) = \log_b(x - h)$, the vertical asymptote is $x = h$.

• Transformations:

  • Horizontal Shift: $f(x) = \log_b(x - h)$ shifts the graph right by $h$ units.

  • Vertical Shift: $f(x) = \log_b(x) + k$ shifts the graph up by $k$ units.

  • Reflection: $f(x) = -\log_b(x)$ reflects the graph over the x-axis.

  • Stretch/Compression: $f(x) = a\log_b(x)$ stretches (if $|a| > 1$) or compresses (if $0 < |a| < 1$) vertically.

• Example: For $f(x) = -5\log_3(x) - 4$:

  • Reflected over x-axis

  • Stretched vertically by factor of 5

  • Shifted down by 4 units

Intercepts of Logarithmic Functions

To find the x-intercept of $f(x) = \log_5(x)$, set $f(x) = 0$:

• $0 = \log_5(x)$

• $x = 1$

Thus, the x-intercept is at $(1, 0)$.

Compound Interest and Exponential Growth

Compound Interest Formula

Compound interest allows you to earn interest on both the principal and the accumulated interest. The general formula for compound interest is:

• $P(t) = P_0 (1 + r/n)^{nt}$

• $P_0$: Initial principal

• $r$: Annual interest rate (as a decimal)

• $n$: Number of compounding periods per year

• $t$: Time in years

Examples of Compounding

• Annually: $n = 1$

• Quarterly: $n = 4$

• Monthly: $n = 12$

• Daily: $n = 365$

Periodic Compounding

As the number of compounding periods increases, the accumulated amount approaches a limit. The formula for periodic compounding is:

• $P(t) = P_0 \left(1 + \frac{r}{n}\right)^{nt}$

For example, compounding monthly ($n = 12$) or daily ($n = 365$) yields slightly higher returns than annual compounding.

Continuous Compounding

When compounding occurs infinitely often, we use the formula for continuous compounding:

• $P(t) = P_0 e^{rt}$

Here, $e$ is Euler's number ($\approx 2.71828$).

Example:

• Invest $1000$at $6\%$ annual interest, compounded continuously for $10$ years:

• $P(10) = 1000 \cdot e^{0.06 \times 10} \approx 1000 \cdot e^{0.6} \approx 1822.12$

Exponential Functions and Applications

General Form of Exponential Functions

An exponential function is defined as:

• $f(x) = ab^x$

• $a$: Initial value

• $b$: Base (if $b > 1$, growth; if $0 < b < 1$, decay)

Exponential Growth and Decay

• Growth: $b > 1$

• Decay: $0 < b < 1$

Exponential models are used in finance, population studies, radioactive decay, and more.

Example: Radioactive Decay

• Suppose a radioactive material decays so that $400$grams remain after $4$ years, modeled by $m(t) = 400e^{-0.06t}$.

• To find the half-life, solve $0.5 = e^{-0.06t}$ for $t$:

$\ln(0.5) = -0.06t$ $t = \frac{\ln(0.5)}{-0.06} \approx 11.55$ years

Solving for Time or Rate

To solve for the time required for an investment to double (or halve), use logarithms:

• For doubling: $2 = e^{rt}$, so $t = \frac{\ln 2}{r}$

• For halving: $0.5 = e^{rt}$, so $t = \frac{\ln 0.5}{r}$

Applications and Practice Problems

Common Applications

• Compound Interest: Calculating future value of investments with different compounding periods.

• Exponential Growth: Modeling population growth, such as bacteria doubling every fixed period.

• Exponential Decay: Modeling radioactive decay or depreciation of assets.

Sample Problems

1. Find the accumulated value of $10,000$ invested for $5$years at $5.5\%$ interest, compounded:

   • Annually: $P = 10,000(1 + 0.055)^5$

   • Quarterly: $P = 10,000\left(1 + \frac{0.055}{4}\right)^{20}$

   • Monthly: $P = 10,000\left(1 + \frac{0.055}{12}\right)^{60}$

   • Continuously: $P = 10,000e^{0.055 \times 5}$

2. Find a model for the minimum wage growth from 1970 to 2000, and use it to estimate the wage in 2023.

3. Find the half-life of a radioactive material given its decay model.

4. Model the population of bacteria doubling every 3 hours and determine when it reaches a certain size.

Table: Compounding Frequency and Accumulated Value

Compounding Frequency	Formula	Example (P0 = $1000$, r = 6%, t = 10 yrs)
Annually	$P = 1000(1 + 0.06)^{10}$	$1790.85$
Quarterly	$P = 1000\left(1 + \frac{0.06}{4}\right)^{40}$	$1814.02$
Monthly	$P = 1000\left(1 + \frac{0.06}{12}\right)^{120}$	$1818.40$
Continuously	$P = 1000e^{0.06 \times 10}$	$1822.12$

Additional info: These notes also include worked examples for finding exponential models from data, solving for unknowns using logarithms, and interpreting real-world scenarios involving exponential growth and decay.