C5

Exponential and Logarithmic Functions

Introduction to Exponential and Logarithmic Functions

Exponential and logarithmic functions are fundamental in College Algebra, providing tools for modeling growth, decay, and many real-world phenomena. Understanding their properties, graphs, and transformations is essential for further study in mathematics and science.

Exponential Functions

Definition and Properties

• Exponential Function: A function of the form $f(x) = b^x$, where $b > 0$ and $b \neq 1$.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• y-intercept: $(0, 1)$

• Horizontal Asymptote: $y = 0$

Example: Find the base $b$ if the graph of $y = b^x$ contains the point $(-7, 1073741825)$.

• Substitute: $1073741825 = b^{-7}$

• Solve for $b$:

$1073741825 = b^{-7}$ $b = (1073741825)^{-1/7}$

Inverse of Exponential Functions

• The inverse of $f(x) = b^x$ is $f^{-1}(x) = \log_b x$.

• To find the inverse, switch $x$ and $y$ and solve for $y$:

$y = b^x \implies x = b^y \implies y = \log_b x$

Logarithmic Functions

Definition and Properties

• Logarithmic Function: $f(x) = \log_b x$, where $b > 0$, $b \neq 1$.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• x-intercept: $(1, 0)$

• Vertical Asymptote: $x = 0$

Example: Which of the following are equal?

• $\log_2 8 = 3$

• $\frac{\ln 8}{\ln 2} = 3$

• $\frac{\log 8}{\log 2} = 3$

All are equal by the change of base formula.

Inverse Relationship

• If $f(x) = b^x$, then $f^{-1}(x) = \log_b x$.

• Exponential and logarithmic functions are inverses of each other.

Graphing Exponential and Logarithmic Functions

• Graphs of $y = b^x$ and $y = \log_b x$ are reflections over the line $y = x$.

• Exponential functions are always increasing for $b > 1$ and decreasing for $0 < b < 1$.

• Logarithmic functions are always increasing for $b > 1$ and decreasing for $0 < b < 1$.

Table: Example Values for $f(x) = 2^x$ and $f^{-1}(x) = \log_2 x$

x	$f(x) = 2^x$	$f^{-1}(x) = \log_2 x$
-2	0.25	-2
-1	0.5	-1
0	1	0
1	2	1
2	4	2

Transformations of Logarithmic Functions

General Form and Transformations

All transformations of the parent logarithmic function $f(x) = \log_b x$ have the form:

$f(x) = a \log_b (x - h) + k$

• Horizontal shift: $h$ units (right if $h > 0$, left if $h < 0$)

• Vertical shift: $k$ units (up if $k > 0$, down if $k < 0$)

• Reflection: Over x-axis if $a < 0$

• Vertical stretch/compression: By factor $|a|$

Example: Describe the transformations of $f(x) = -2 \log_2 (x - 9) + 2$ from $f(x) = \log_2 x$.

• Shift right 9 units

• Vertical stretch by 2

• Reflect over x-axis

• Shift up 2 units

Graphical Transformations

Function	Transformation
$f(x) = \log_2 (x-3)$	Shift 3 units right
$f(x) = \log_2 (-x)$	Reflect over y-axis
$f(x) = \log_2 (x+4)$	Shift 4 units left
$f(x) = -\log_2 (x-3)$	Reflect over x-axis, shift 3 units right
$f(x) = 2\log_2 (x)$	Vertical stretch by 2
$f(x) = 3\log_2 (x)$	Vertical stretch by 3

Domain, Range, Intercepts, and Asymptotes

Finding Key Features

• Domain: Set the argument of the logarithm greater than zero and solve for $x$.

• Range: All real numbers for logarithmic functions.

• x-intercept: Set $f(x) = 0$ and solve for $x$.

• y-intercept: Set $x = 0$ and solve for $f(0)$ (if in domain).

• Vertical asymptote: Set the argument of the logarithm equal to zero and solve for $x$.

Example: For $f(x) = \log_2 (x+9)$:

• Domain: $x + 9 > 0 \implies x > -9$

• Range: $(-\infty, \infty)$

• x-intercept: $x = -8$

• Vertical asymptote: $x = -9$

Inverse Functions

Finding the Inverse

• Switch $x$ and $y$ in the equation and solve for $y$.

• The inverse of a logarithmic function is an exponential function.

Example: For $f(x) = \log_2 (x+9)$, the inverse is $f^{-1}(x) = 2^x - 9$.

Practice Problems

Sample Problems and Solutions

1. For $f(x) = 2^x - 4$, find:

   • Domain: $(-\infty, \infty)$

   • Range: $(-4, \infty)$

   • y-intercept: $(0, -3)$

   • x-intercept: $x = \log_2 4 = 2$

   • Equation of asymptote: $y = -4$

   • Sketch: Exponential curve shifted down 4 units

2. For $f(x) = \log_2 (x+9) - 4$, find:

   • Domain: $x > -9$

   • Range: $(-\infty, \infty)$

   • x-intercept: $x = -8$

   • Vertical asymptote: $x = -9$

   • Inverse: $f^{-1}(x) = 2^{x+4} - 9$

   • Sketch: Logarithmic curve shifted left 9 units and down 4 units

Summary Table: Exponential vs. Logarithmic Functions

Feature	Exponential $y = b^x$	Logarithmic $y = \log_b x$
Domain	$(-\infty, \infty)$	$(0, \infty)$
Range	$(0, \infty)$	$(-\infty, \infty)$
y-intercept	$(0, 1)$	None
x-intercept	None	$(1, 0)$
Asymptote	$y = 0$	$x = 0$
Inverse	$y = \log_b x$	$y = b^x$

Additional info: The notes emphasize the graphical relationship between exponential and logarithmic functions, their transformations, and how to find and interpret their inverses. Practice problems reinforce the identification of domain, range, intercepts, and asymptotes, as well as graphing skills.