Exponential and Logarithmic Functions: Core Concepts and Applications

Chapter 4: Exponential and Logarithmic Functions

Exponential Rules

Understanding the rules of exponents is essential for manipulating exponential expressions and solving equations involving exponents. These rules are foundational for college algebra and are prerequisites for advanced topics.

• Product Rule: $a^n \cdot a^m = a^{n+m}$ Example: $2^3 \cdot 2^4 = 2^{3+4} = 2^7 = 128$

• Product Rule (Different Bases, Same Exponent): $a^n \cdot b^n = (a \cdot b)^n$ Example: $3^2 \cdot 4^2 = (3 \cdot 4)^2 = 12^2 = 144$

• Quotient Rule: $a^n \div a^m = a^{n-m}$ Example: $2^8 \div 2^5 = 2^{8-5} = 2^3 = 8$

• Quotient Rule (Different Bases, Same Exponent): $a^n \div b^n = \left(\frac{a}{b}\right)^n$ Example: $4^2 \div 3^2 = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$

• Power Rule: $(b^n)^m = b^{n \cdot m}$ Example: $(2^2)^3 = 2^{2 \cdot 3} = 2^6 = 64$

• Fractional Exponents: $b^{1/n} = \sqrt[n]{b}$ Example: $8^{1/3} = \sqrt[3]{8} = 2$

• Fractional Exponents (General): $b^{m/n} = (\sqrt[n]{b})^m$ Example: $27^{2/3} = (\sqrt[3]{27})^2 = 3^2 = 9$

• Negative Exponent Rule: $b^{-n} = \frac{1}{b^n}$ Example: $2^{-2} = \frac{1}{2^2} = 0.25$

• Zero Exponent Rule: $b^0 = 1$ Example: $7^0 = 1$

• One Exponent Rule: $b^1 = b$ Example: $5^1 = 5$

• One to Any Power: $1^n = 1$ Example: $1^8 = 1$

Exponential Functions

An exponential function is defined as $f(x) = b^x$, where $b$ is a positive constant other than 1 ($b > 0$, $b \neq 1$), and $x$ is any real number. The base $b$ must be constant and positive, and the exponent is the variable.

• Examples of Exponential Functions:

  • $f(x) = 2^x$ (base 2)

  • $f(x) = 3^{x+1}$ (base 3)

  • $f(x) = (1/4)^x$ (base $1/4$)

• Non-Exponential Functions: Functions where the base is variable or negative, such as $f(x) = x^2$, $g(x) = 1^x$, or $f(x) = x^x$, are not exponential functions.

Applications of Exponential Functions

• Population growth

• Spread of pandemics

• Radioactive decay

• Any process involving rapid increase or decrease

Evaluating Exponential Functions

• To evaluate $f(x) = b^x$ for a given $x$, substitute the value and compute using a calculator if necessary.

• Example: $f(x) = 3.2^x$ for $x = 5$

Graphing Exponential Functions

The graph of an exponential function depends on the value of the base $b$:

• If $b > 1$, the function is increasing.

• If $0 < b < 1$, the function is decreasing.

• The y-intercept is always at $(0, 1)$.

• There is no x-intercept.

• The horizontal asymptote is $y = 0$.

• The domain is $x \in (-\infty, \infty)$; the range is $y \in (0, \infty)$.

Comparing Increasing and Decreasing Exponential Functions

• For $f(x) = b^x$ with $b > 1$, the graph rises rapidly as $x$ increases.

• For $f(x) = b^x$ with $0 < b < 1$, the graph falls rapidly as $x$ increases.

Effect of the Base on Steepness

• The larger the base $b$, the steeper the increase for $b > 1$.

• The smaller the base $b$ (but still $0 < b < 1$), the steeper the decrease.

The Number $e$

The mathematical constant $e$ is defined as the value that $\left(1 + \frac{1}{n}\right)^n$ approaches as $n$ becomes very large. Rounded to five decimal places, $e \approx 2.71828$.

• $e$ is the base of the natural exponential function $f(x) = e^x$.

• It is used extensively in continuous growth and decay models.

Graph of $y = e^x$

• Domain: $x \in (-\infty, \infty)$

• Range: $y \in (0, \infty)$

• Horizontal asymptote: $y = 0$

• Key points: $(0, 1)$, $(1, e)$, $(2, e^2)$

Compound Interest

Compound interest is a common application of exponential functions in finance. The formula for compound interest is:

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

• Where:

  • $A$ = final amount (future value)

  • $P$ = initial amount (present value)

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

  • $n$ = number of compounding periods per year

  • $t$ = time in years

Compounding Period	n
Annually	1
Semiannually	2
Quarterly	4
Monthly	12
Weekly	52
Daily	365

Continuous Compounding

When interest is compounded continuously, the formula is:

• $A = Pe^{rt}$

• Where $e$ is the mathematical constant described above.

Example: Compound Interest Calculation

• Amy deposits $10,000 at 9% annual interest, compounded monthly, for 5 years:

• $r = 0.09$, $n = 12$, $t = 5$

• $A = 10,000 \left(1 + \frac{0.09}{12}\right)^{12 \cdot 5}$

• Balance after 5 years: $15,656.81$

Example: Continuous Compounding Calculation

• $2,340$ deposited at 3.1% annual interest, compounded continuously, for 3 years:

• $A = 2340 \cdot e^{0.031 \cdot 3} \approx 2,568.06$

Summary Table: Key Properties of Exponential Functions

Property	Exponential Growth ($b > 1$)	Exponential Decay ($0 < b < 1$)
Domain	$(-\infty, \infty)$	$(-\infty, \infty)$
Range	$(0, \infty)$	$(0, \infty)$
y-intercept	$(0, 1)$	$(0, 1)$
x-intercept	None	None
Horizontal Asymptote	$y = 0$	$y = 0$
Behavior as $x \to \infty$	$y \to \infty$	$y \to 0$
Behavior as $x \to -\infty$	$y \to 0$	$y \to \infty$

Additional info: The notes above include expanded explanations, definitions, and examples to ensure a self-contained and comprehensive study guide for college algebra students.