Chapter 3: Exponential and Logarithmic Functions – Precalculus Study Notes

Chapter 3: Exponential and Logarithmic Functions

3.1 Exponential Functions

Exponential functions are fundamental in modeling growth and decay processes. They are defined by a constant base raised to a variable exponent.

• Definition: An exponential function with base b is defined by $f(x) = b^x$ or $y = b^x$, where $b > 0$ and $b \neq 1$.

• Domain: All real numbers $(-\infty, \infty)$.

• Range: All positive real numbers $(0, \infty)$.

• Y-intercept: $(0, 1)$ since $b^0 = 1$.

• Asymptote: The x-axis ($y = 0$) is a horizontal asymptote.

• Growth/Decay: If $b > 1$, the function is increasing (exponential growth). If $0 < b < 1$, the function is decreasing (exponential decay).

Example: $f(x) = 42.2(1.56)^x$ models the average amount spent after $x$ hours. $f(3) = 42.2(1.56)^3 = 140.21$

Graphing Exponential Functions

• Plot points for several values of $x$ to observe the rapid increase or decrease.

• Exponential functions never touch the x-axis.

Transformations of Exponential Functions

Transformation	Equation	Description
Vertical shift	$g(x) = b^x + k$	Shifts graph up/down $k$ units
Horizontal shift	$g(x) = b^{x-h}$	Shifts graph right/left $h$ units
Reflection	$g(x) = -b^x$	Reflects across x-axis
Vertical stretch/shrink	$g(x) = a b^x$	Stretches if $|a| > 1$, shrinks if $0 < |a| < 1$

Exponential Functions with Base $e$

• The number $e \approx 2.71828$ is the natural base.

• The function $f(x) = e^x$ is called the natural exponential function.

Compound Interest Formulas

• n compounding periods per year: $A = P\left(1 + \frac{r}{n}\right)^{nt}$

• Continuous compounding: $A = Pe^{rt}$

• Example: $10,000$ invested at $1.08\%$ compounded quarterly for $5$years:$A = 10,554.08$

3.2 Logarithmic Functions

Logarithmic functions are the inverses of exponential functions and are used to solve for exponents.

• Definition: For $x > 0$, $b > 0$, $b \neq 1$, $y = \log_b x$ is equivalent to $b^y = x$.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Y-intercept: None

• X-intercept: $(1, 0)$

• Asymptote: The y-axis ($x = 0$) is a vertical asymptote.

Example: $\log_2 8 = 3$ because $2^3 = 8$

Changing Forms

• Exponential to Logarithmic: $b^y = x \rightarrow y = \log_b x$

• Logarithmic to Exponential: $y = \log_b x \rightarrow b^y = x$

Common and Natural Logarithms

• Common logarithm: $\log_{10} x$ or $\log x$

• Natural logarithm: $\log_e x$ or $\ln x$

Properties of Logarithms

Property	General	Common	Natural
Identity	$\log_b 1 = 0$	$\log 1 = 0$	$\ln 1 = 0$
Base	$\log_b b = 1$	$\log 10 = 1$	$\ln e = 1$
Inverse	$\log_b b^x = x$	$\log 10^x = x$	$\ln e^x = x$
Exponent	$b^{\log_b x} = x$	$10^{\log x} = x$	$e^{\ln x} = x$

3.3 Properties of Logarithms

Logarithmic properties allow for the simplification, expansion, and condensation of logarithmic expressions.

• Product Rule: $\log_b (MN) = \log_b M + \log_b N$

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

• Power Rule: $\log_b (M^p) = p \log_b M$

Example: $\log_2 (x^4 y^5) = 4\log_2 x + 5\log_2 y$

Expanding and Condensing Logarithmic Expressions

• Expand: Use product, quotient, and power rules to write as a sum/difference of logs.

• Condense: Combine multiple logs into a single logarithm.

Change-of-Base Property

• $\log_b M = \frac{\log_a M}{\log_a b}$

• Common: $\log_b M = \frac{\log M}{\log b}$

• Natural: $\log_b M = \frac{\ln M}{\ln b}$

3.4 Exponential and Logarithmic Equations

Solving exponential and logarithmic equations is essential for applications in science, finance, and engineering.

• Solving Exponential Equations:

  1. Express both sides with the same base if possible, then set exponents equal.

  2. If not possible, take logarithms of both sides and solve for the variable.

• Solving Logarithmic Equations:

  1. Combine logs into a single logarithm if possible.

  2. Rewrite in exponential form and solve for the variable.

  3. Check for extraneous solutions (argument of log must be positive).

• One-to-One Property: If $\log_b M = \log_b N$, then $M = N$.

Example: $2^x = 8 \implies x = 3$; $\log_2 (x-3) = 4 \implies x-3 = 16 \implies x = 19$

3.5 Exponential Growth and Decay; Modeling Data

Exponential and logarithmic models are widely used to describe real-world phenomena such as population growth, radioactive decay, and temperature change.

Exponential Growth and Decay Models

• General Model: $A(t) = A_0 e^{kt}$, where $A_0$ is the initial amount, $k$ is the growth ($k > 0$) or decay ($k < 0$) rate, and $t$ is time.

• Exponential Growth: $k > 0$

• Exponential Decay: $k < 0$

Example: Population growth: $A(t) = 807 e^{0.02529 t}$

Logistic Growth Model

• Model: $f(t) = \frac{c}{1 + a e^{-b t}}$ or $f(t) = \frac{c}{1 + a b^{-t}}$

• Used for populations with limiting factors (e.g., carrying capacity).

Newton's Law of Cooling

• Model: $T = C + (T_0 - C) e^{kt}$, where $C$ is the surrounding temperature, $T_0$ is the initial temperature, and $k$ is a negative constant.

Example: Cooling from $100^\circ$C to $35^\circ$C in a room at $30^\circ$C.

Expressing Exponential Models in Base $e$

• Any exponential function $y = ab^x$ can be rewritten as $y = a e^{kx}$, where $k = \ln b$.

Example: $y = 4(2.7)^x = 4e^{x}$ (since $\ln 2.7 \approx 1$)

Additional info: These notes are based on class and textbook materials for a standard Precalculus course, focusing on Chapter 3: Exponential and Logarithmic Functions. All key properties, formulas, and example problems are included for comprehensive exam preparation.