Logarithmic Functions and Their Properties in Business Calculus

Part I: Logarithmic Function

Introduction to Exponential and Logarithmic Functions

Exponential and logarithmic functions are fundamental in business calculus, especially for modeling growth, decay, and solving equations involving exponents. The exponential function $f(x) = a^x$ (where $a > 0$, $a \neq 1$) is one-to-one and thus has an inverse, the logarithmic function.

• Exponential Function: $f(x) = a^x$

• Inverse Function: $f^{-1}(x) = \log_a x$

• Cancellation Rules:

  • $f^{-1}(f(x)) = x$ for all $x$ in the domain of $f$

  • $f(f^{-1}(x)) = x$ for all $x$ in the domain of $f^{-1}$

Properties of Inverse Functions

The inverse of the exponential function $f(x) = a^x$ is called the logarithmic function to the base a and is denoted $f^{-1}(x) = \log_a x$.

• General Property: $f(x) = y \iff f^{-1}(y) = x$

• For exponentials and logarithms: $a^x = y \iff \log_a y = x$

• Domains and Ranges:

  • Domain of $\log_a x$: $(0, \infty)$

  • Range of $\log_a x$: $(-\infty, \infty)$

  • Domain of $a^x$: $(-\infty, \infty)$

  • Range of $a^x$: $(0, \infty)$

Cancellation Rules for Logarithms and Exponentials

Rule	Condition
$\log_a a^x = x$	for all real $x$
$a^{\log_a x} = x$	for $x > 0$

Converting Between Exponential and Logarithmic Forms

• Exponential to Logarithmic: $a^x = y \implies x = \log_a y$

• Logarithmic to Exponential: $\log_a x = y \implies x = a^y$

These conversions are essential for solving equations involving exponents or logarithms.

Examples: Converting Forms and Solving Equations

• Convert $\log_2 \frac{1}{16} = -4$ to exponential form: $2^{-4} = \frac{1}{16}$

• Solve $\log_3 (3x-1) = 2$: $3x-1 = 3^2 = 9 \implies 3x = 10 \implies x = \frac{10}{3}$

• Solve $e^{x/2} = 5$: $\log_e e^{x/2} = \log_e 5 \implies \frac{x}{2} = \log_e 5 \implies x = 2 \log_e 5$

Graphing Exponential and Logarithmic Functions

Graphs and Key Features

Exponential and logarithmic functions are reflections of each other across the line $y = x$.

$f(x) = a^x$	$f^{-1}(x) = \log_a x$
Domain: $(-\infty, \infty)$	Domain: $(0, \infty)$
Range: $(0, \infty)$	Range: $(-\infty, \infty)$
Horizontal Asymptote: $y=0$	Vertical Asymptote: $x=0$
Points: $(-1, 1/a), (0, 1), (1, a)$	Points: $(1/a, -1), (1, 0), (a, 1)$
Increasing if $a > 1$; decreasing if $0 < a < 1$	Increasing if $a > 1$; decreasing if $0 < a < 1$

Example: Graphs

• Graph $f(x) = 2^x$ and $f^{-1}(x) = \log_2 x$: The exponential curve rises rapidly, while the logarithmic curve increases slowly and passes through $(1,0)$.

• Graph $f(x) = (1/2)^x$ and $f^{-1}(x) = \log_{1/2} x$: The exponential curve decreases, and the logarithmic curve is decreasing as $x$ increases.

Properties and Laws of Logarithms

Definition and Interpretation

The logarithm $\log_a x$ is the power to which the base $a$ must be raised to produce $x$.

• Example: $\log_2 16 = 4$ because $2^4 = 16$

• Example: $\log_5 \frac{1}{5} = -1$ because $5^{-1} = \frac{1}{5}$

Special Logarithms

• Natural Logarithm: $\log_e x = \ln x$

• Common Logarithm: $\log_{10} x = \log x$

Calculators can evaluate base $e$ or base $10$logarithms. In applications,$\log x$often means$\ln x$ unless otherwise specified.

Logarithm Laws

• $\log_a (xy) = \log_a x + \log_a y$

• $\log_a \left(\frac{x}{y}\right) = \log_a x - \log_a y$

• $\log_a x^r = r \log_a x$

• $\log_a a = 1$

• $\log_a 1 = 0$

Identities (Cancellation Rules)

Identity	Condition
$\log_a a^x = x$	for all real $x$
$a^{\log_a x} = x$	for $x > 0$

Change of Base Formula

• If $a, b, x$ are positive with $a \neq 1$ and $b \neq 1$, then: $\log_a x = \frac{\log_b x}{\log_b a}$

• Example: $\log_{19} 2 = \frac{\ln 2}{\ln 19}$

Solving Logarithmic and Exponential Equations

Solving Logarithmic Equations

1. Isolate the logarithm on one side or write as a single logarithm on each side.

2. Compose the exponential function with the same base as the logarithm to both sides and simplify.

3. Solve for the variable.

4. Check each proposed solution with the domain of the original equation.

• Example: Solve $\log x + \log(x+1) = \log 12$ $\log(x(x+1)) = \log 12 \implies x(x+1) = 12 \implies x^2 + x - 12 = 0$ $x = 3$ or $x = -4$ (but $x = -4$ is not in the domain, so $x = 3$ is the solution)

Solving Exponential Equations

1. Reduce the equation to one of the forms: $a^{f(x)} = b$, $a^{f(x)} = a^{g(x)}$, or $a^{f(x)} = b^{g(x)}$.

2. Compose a logarithmic function to both sides.

3. Simplify and solve for the variable.

• Example: Solve $4^x = 32$ $4^x = 2^5 \implies (2^2)^x = 2^5 \implies 2^{2x} = 2^5 \implies 2x = 5 \implies x = \frac{5}{2}$

• Example: Solve $500e^{-x} = 300$ $e^{-x} = \frac{3}{5} \implies -x = \ln \frac{3}{5} \implies x = -\ln \frac{3}{5}$

Summary Table: Key Properties of Exponential and Logarithmic Functions

Property	Exponential $f(x) = a^x$	Logarithmic $f^{-1}(x) = \log_a x$
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
Asymptote	Horizontal: $y=0$	Vertical: $x=0$
Key Points	$(-1, 1/a), (0, 1), (1, a)$	$(1/a, -1), (1, 0), (a, 1)$
Monotonicity	Increasing if $a > 1$; decreasing if $0 < a < 1$	Increasing if $a > 1$; decreasing if $0 < a < 1$

Additional info:

• Logarithmic and exponential functions are widely used in business for modeling compound interest, population growth, and decay processes.

• Understanding the properties and manipulation of logarithms is essential for solving real-world business calculus problems.