Exponential and Logarithmic Functions: Properties, Graphs, and Transformations

Exponential Functions

Definition and Basic Properties

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. Exponential functions are fundamental in modeling growth and decay in various fields such as biology, finance, and physics.

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

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

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

• No X-intercept: The graph never crosses the x-axis.

• One-to-one function: Each $x$ value corresponds to a unique $y$ value, and vice versa.

• Horizontal asymptote: $y = 0$

• If $b > 1$, $f(x) = b^x$ is increasing (rises to the right).

• If $0 < b < 1$, $f(x) = b^x$ is decreasing (falls to the right).

Example: $f(x) = 2^x$ is increasing; $f(x) = (1/2)^x$ is decreasing.

Graphing Exponential Functions

To graph an exponential function, plot several points by substituting values for $x$ and calculate $f(x)$. Identify the horizontal asymptote and the y-intercept.

• For transformations such as $f(x) = b^{x-c} + d$, shift the graph horizontally by $c$ units and vertically by $d$ units.

Logarithmic Functions

Definition and Basic Properties

A logarithmic function with base $b$ is defined as $f(x) = \log_b x$, where $b > 0$ and $b \neq 1$. The logarithmic function is the inverse of the exponential function.

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

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

• X-intercept: $(1, 0)$ since $\log_b 1 = 0$

• No Y-intercept: The graph never crosses the y-axis.

• Vertical asymptote: $x = 0$

• If $b > 1$, $f(x) = \log_b x$ is increasing.

Inverse Properties:

• $\log_b b^x = x$

• $b^{\log_b x} = x$

Common and Natural Logarithms

• Common logarithm: Base 10, written as $\log x$

• Natural logarithm: Base $e$, written as $\ln x$

Properties of Logarithms

• 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: $\log_b M^p = p \log_b M$

Transformations of Logarithmic Functions

Logarithmic functions can be shifted, reflected, stretched, or shrunk using transformations:

Transformation	Equation	Description
Vertical translation	$g(x) = \log_b x + c$ $g(x) = \log_b x - c$	Shifts graph up/down $c$ units
Horizontal translation	$g(x) = \log_b(x + c)$ $g(x) = \log_b(x - c)$	Shifts graph left/right $c$ units; vertical asymptote at $x = -c$ or $x = c$
Reflection	$g(x) = -\log_b x$ $g(x) = \log_b(-x)$	Reflects graph about x-axis or y-axis
Vertical stretching/shrinking	$g(x) = c \log_b x$	Stretches if $c > 1$, shrinks if $0 < c < 1$
Horizontal stretching/shrinking	$g(x) = \log_b(cx)$	Shrinks if $c > 1$, stretches if $0 < c < 1$

Solving Logarithmic Equations Using the One-to-One Property

To solve equations involving logarithms, use the one-to-one property:

• Express the equation in the form $\log_b M = \log_b N$.

• By the one-to-one property, $M = N$.

Summary Table: Key Properties of Exponential and Logarithmic Functions

Function	Domain	Range	Intercepts	Asymptote
$f(x) = b^x$	$(-\infty, \infty)$	$(0, \infty)$	Y: $(0,1)$	$y = 0$
$f(x) = \log_b x$	$(0, \infty)$	$(-\infty, \infty)$	X: $(1,0)$	$x = 0$