8.3 Exponential & Logarithmic Functions, Laws & Modeling

Exponential & Logarithmic Functions, Laws & Modeling

Core Advanced Algebra: Exponential and logarithmic relationships model multiplicative growth, decay dynamics, and inverse scaling across advanced mathematics. The ACCUPLACER Advanced Algebra and Functions test measures your proficiency in converting between exponential and logarithmic representations, applying logarithmic operational identities, solving exponential equations with like and unlike bases, filtering extraneous roots, and evaluating real-world growth and decay models.

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Exponential Functions: Structural Properties, Growth & Decay

An exponential function is a non-linear function where the independent variable $x$ appears in the exponent:

$$f(x) = a \cdot b^x \quad (a \neq 0, \; b > 0, \; b \neq 1)$$

• $a$ is the initial value (since $f(0) = a \cdot b^0 = a$, corresponding to the $y$-intercept $(0, a)$).
• $b$ is the base (growth or decay factor).

           EXPONENTIAL GROWTH (b > 1)                     EXPONENTIAL DECAY (0 < b < 1)
      y ▲                                             y ▲
        │          ╭───                                 │   ───╮
        │         ╭╯                                    │      ╰╮
        │        ╭╯                                     │       ╰╮
        │       ╭╯                                      │        ╰╮
        │     (0,a)                                     │       (0,a)
        │   ───╯                                        │         ╰───
        └────────────────────────► x                    └────────────────────────► x
        Horizontal Asymptote: y = 0                     Horizontal Asymptote: y = 0

Key Graphical Characteristics of $f(x) = a \cdot b^x$ ($a > 0$)

1. Domain: $(-\infty, \infty)$ (all real numbers).
2. Range: $(0, \infty)$ (strictly positive real numbers).
3. Horizontal Asymptote: The line $y = 0$ ($x$-axis).
4. Monotonicity:
   • If $b > 1$, the function models Exponential Growth (strictly increasing). As $x \to \infty, f(x) \to \infty$; as $x \to -\infty, f(x) \to 0$.
   • If $0 < b < 1$, the function models Exponential Decay (strictly decreasing). As $x \to \infty, f(x) \to 0$; as $x \to -\infty, f(x) \to \infty$.
5. Growth and Decay Rates ($r$):
   • Growth Factor: $b = 1 + r$ (where $r > 0$ is the percentage growth rate in decimal form).
   • Decay Factor: $b = 1 - r$ (where $0 < r < 1$ is the percentage decay rate in decimal form).

The Natural Exponential Base $e$

The mathematical constant $e$ (Euler's number) is an irrational constant defined as the limiting value:

$$\ne = \lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n \approx 2.718281828\dots$$

The natural exponential function $f(x) = e^x$ is the foundational model for continuous compound interest, unrestricted population dynamics, and radioactive decay.

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Logarithmic Functions: Formal Definition & Inverse Properties

A logarithm represents the exponent to which a base must be raised to produce a specified value. Logarithms are the exact algebraic inverses of exponential functions.

The Fundamental Equivalence Identity

$$\log_b(x) = y \iff b^y = x \quad (b > 0, \; b \neq 1, \; x > 0)$$

• Exponential form: $2^5 = 32 \iff$ Logarithmic form: $\log_2(32) = 5$
• Exponential form: $10^{-3} = 0.001 \iff$ Logarithmic form: $\log_{10}(0.001) = -3$
• Exponential form: $e^k = M \iff$ Logarithmic form: $\ln(M) = k$

Domain, Range & Asymptotes of Logarithmic Functions

Because logarithmic functions are the inverses of exponential functions, their domains and ranges are completely reversed:

• Domain: $(0, \infty)$ — The argument of a logarithm must be strictly positive. $\log_b(0)$ and $\log_b(\text{negative})$ are undefined in the real number system.
• Range: $(-\infty, \infty)$ (all real numbers).
• Vertical Asymptote: The vertical line $x = 0$ ($y$-axis).
• $x$-Intercept: $(1, 0)$ because $\log_b(1) = 0$ for any valid base $b$.

Common Logarithms vs. Natural Logarithms

• Common Logarithm (Base 10): Written as $\log(x) \equiv \log_{10}(x)$.
• Natural Logarithm (Base $e$): Written as $\ln(x) \equiv \log_e(x)$.

Inverse Cancellation Identities

1. $\log_b(b^x) = x$ for all $x \in \mathbb{R}$, and $b^{\log_b x} = x$ for all $x > 0$
2. $\ln(e^x) = x$ for all $x \in \mathbb{R}$, and $e^{\ln x} = x$ for all $x > 0$
3. $\log_b(b) = 1$ and $\ln(e) = 1$
4. $\log_b(1) = 0$ and $\ln(1) = 0$

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Fundamental Laws of Logarithms & Change of Base

Logarithmic properties convert complex multiplication, division, and exponentiation into simpler arithmetic operations.

Logarithm Laws Reference Table

Expanding and Condensing Logarithmic Expressions

• Expanding: Break down composite arguments into sums and differences of single-variable logarithms: $$\log_b\left(\frac{x^4 \sqrt{y}}{z^3}\right) = \log_b(x^4) + \log_b(y^{1/2}) - \log_b(z^3) = 4\log_b x + \frac{1}{2}\log_b y - 3\log_b z$$
• Condensing: Combine multiple logarithmic terms with identical bases into a single compact logarithmic term: $$3\ln(x) - 2\ln(y) + \frac{1}{2}\ln(w) = \ln(x^3) - \ln(y^2) + \ln(\sqrt{w}) = \ln\left(\frac{x^3 \sqrt{w}}{y^2}\right)$$

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Solving Exponential Equations

Method 1: Equating Common Bases (One-to-One Property)

When both sides of an exponential equation can be expressed as powers of the same base $b$:

$$b^{u(x)} = b^{v(x)} \iff u(x) = v(x)$$

Worked Example: Equating Common Bases

Solve for $x$: $27^{2x - 1} = 9^{x + 4}$.

• Step 1 (Express both sides with base 3): $27 = 3^3$ and $9 = 3^2$. $$(3^3)^{2x - 1} = (3^2)^{x + 4} \implies 3^{3(2x - 1)} = 3^{2(x + 4)}$$
• Step 2 (Set exponents equal): $$3(2x - 1) = 2(x + 4) \implies 6x - 3 = 2x + 8$$
• Step 3 (Solve for $x$): $$4x = 11 \implies x = \frac{11}{4}$$

Method 2: Solving Equations with Unlike Bases Using Logarithms

When bases cannot be matched, isolate the exponential expression and take the natural logarithm ($\ln$) or common logarithm ($\log$) of both sides.

Worked Example: Unlike Bases

Solve for $x$: $5 \cdot 2^{3x - 2} = 80$.

• Step 1 (Isolate exponential term): Divide by 5: $2^{3x - 2} = 16$.
• Step 2 (Recognize base or apply logs): Since $16 = 2^4$: $$3x - 2 = 4 \implies 3x = 6 \implies x = 2$$
• General Case with non-matching base (e.g., $2^{3x-2} = 15$): $$\ln(2^{3x - 2}) = \ln(15) \implies (3x - 2)\ln 2 = \ln 15 \implies 3x - 2 = \frac{\ln 15}{\ln 2} \implies x = \frac{1}{3}\left(\frac{\ln 15}{\ln 2} + 2\right)$$

Method 3: Quadratic Form Exponential Equations

Equations of the form $A(e^x)^2 + B(e^x) + C = 0$ can be solved by substituting $u = e^x$:

$$\ne^{2x} - 5e^x + 6 = 0 \implies (e^x - 2)(e^x - 3) = 0 \implies e^x = 2 \;\text{ or }\; e^x = 3 \implies x = \ln 2 \;\text{ or }\; x = \ln 3$$

(Note: If a factor yields $e^x = -4$, discard it immediately because $e^x > 0$ for all real $x$.)

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Solving Logarithmic Equations & Filtering Extraneous Solutions

Solving Single & Multi-Logarithm Equations

1. Condense all logarithmic terms on each side into a single logarithm using the Product and Quotient rules.
2. Convert to exponential form ($b^y = x$) or apply the one-to-one property ($\log_b u = \log_b v \implies u = v$).
3. Solve the resulting algebraic equation.
4. Check all solutions in original expressions: Discard any root that results in the logarithm of a non-positive number ($\le 0$).

Worked Example: Logarithmic Equation with Extraneous Root

Solve for $x$: $\log_3(x + 6) + \log_3(x) = 3$.

• Step 1 (Condense using Product Rule): $$\log_3(x(x + 6)) = 3$$
• Step 2 (Convert to exponential form): $$x(x + 6) = 3^3 = 27$$
• Step 3 (Form and solve quadratic equation): $$x^2 + 6x - 27 = 0 \implies (x + 9)(x - 3) = 0 \implies x = -9 \quad \text{or} \quad x = 3$$
• Step 4 (Test candidate solutions in original equation):
  • Test $x = 3$: $\log_3(3 + 6) + \log_3(3) = \log_3(9) + \log_3(3) = 2 + 1 = 3$ (Valid!).
  • Test $x = -9$: $\log_3(-9)$ is undefined because the argument $-9 \le 0$. Discard $x = -9$ as an extraneous root.
• Conclusion: The unique valid solution is $x = 3$.

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Real-World Mathematical Modeling: Growth, Decay & Compound Interest

Exponential functions accurately model phenomena where rate of change is proportional to current magnitude.

Applied Exponential Models Reference Table

Compounding Frequencies ($n$)

• Annual: $n = 1$
• Semi-annual: $n = 2$
• Quarterly: $n = 4$
• Monthly: $n = 12$
• Daily: $n = 365$

Worked Example: Continuous Financial Doubling Time

How many years will it take for an investment of $\text{\textdollar}5,000$ to double if it is invested in an account earning $8\%$ annual interest compounded continuously?

• Step 1 (Set up model): $A(t) = P e^{rt} \implies 10,000 = 5,000 e^{0.08 t}$.
• Step 2 (Isolate exponential term): $e^{0.08 t} = 2$.
• Step 3 (Apply natural logarithm): $\ln(e^{0.08 t}) = \ln(2) \implies 0.08 t = \ln(2)$.
• Step 4 (Solve for $t$): $t = \frac{\ln 2}{0.08} \approx \frac{0.69315}{0.08} \approx 8.66\text{ years}$.