Inverse, Exponential, and Logarithmic Functions: Study Notes

Inverse, Exponential, and Logarithmic Functions

Introduction to Exponential Functions

Exponential functions are a fundamental class of functions in algebra where the variable appears in the exponent. These functions model rapid growth or decay and are widely used in science, finance, and engineering.

• Definition: An exponential function has the form $F(x) = a^x$, where $a > 0$ and $a \neq 1$.

• Key Feature: The base $a$ is a constant, and the exponent is the variable $x$.

• Contrast: In a polynomial function, the variable is the base and the exponent is a constant.

Polynomial Function	Exponential Function
$F(x) = x^n$	$F(x) = a^x$

Example: $F(x) = 2^x$ is exponential; $F(x) = x^2$ is polynomial.

Graphs of Exponential Functions

The graph of an exponential function $F(x) = a^x$ depends on the value of $a$:

• If $a > 1$, the function increases rapidly as $x$ increases.

• If $0 < a < 1$, the function decreases as $x$ increases.

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Horizontal Asymptote: $y = 0$

Transformations of Exponential Graphs

Exponential graphs can be transformed by shifting, reflecting, stretching, or compressing the parent function $F(x) = a^x$.

• General Transformation: $g(x) = a^{x-h} + k$

• 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 the $x$-axis if multiplied by $-1$

Graphing Exponential Functions: Examples and Practice

To graph a transformed exponential function, identify the parent function, apply shifts and reflections, and plot key points.

• Example: Graph $g(x) = 2^{x-1} + 3$ as a transformation of $F(x) = 2^x$.

• Practice: Graph $g(x) = -3^{x} + 1$ and match to its graph.

The Number e

The number $e$ is an important mathematical constant approximately equal to 2.71828. Exponential functions with base $e$ are called natural exponential functions.

• Definition: $F(x) = e^x$

• Applications: Compound interest, population growth, radioactive decay

Graphing Exponential Functions with Base e

Graphing $F(x) = e^x$ follows the same principles as other exponential functions. The domain is all real numbers, and the range is positive real numbers.

• Identify the horizontal asymptote at $y = 0$.

• Plot key points such as $(0,1)$ and $(1,e)$.

Introduction to Logarithms

Logarithms are the inverse operations of exponentiation. The logarithm of a number is the exponent to which the base must be raised to produce that number.

• Definition: $y = \log_b x$ means $b^y = x$

• Common Bases: $b = 10$ (common log), $b = e$ (natural log, written $\ln x$)

Exponential Form	Logarithmic Form
$b^y = x$	$y = \log_b x$

Evaluating Logarithms and Properties

Logarithms can be evaluated using properties that relate them to exponents.

• Product Rule: $\log_b (MN) = \log_b M + \log_b N$

• Quotient Rule: $\log_b (M/N) = \log_b M - \log_b N$

• Power Rule: $\log_b (M^k) = k \log_b M$

• Change of Base: $\log_b a = \frac{\log_c a}{\log_c b}$

Graphs of Logarithmic Functions

The graph of a logarithmic function $F(x) = \log_b x$ is the inverse of the exponential function $F(x) = b^x$.

• Domain: $(0, \infty)$

• Range: $(-\infty, \infty)$

• Vertical Asymptote: $x = 0$

• Graph increases if $b > 1$, decreases if $0 < b < 1$

Transformations of Logarithmic Graphs

Logarithmic graphs can be shifted, reflected, and stretched using transformations similar to those for exponential functions.

• General Transformation: $g(x) = \log_b (x-h) + k$

• Apply horizontal and vertical shifts, and reflections as needed.

Properties of Logarithms: Product, Quotient, and Power Rules

Properties of logarithms allow for the expansion and condensation of logarithmic expressions, which is useful for solving equations and simplifying expressions.

Property	Rule	Description
Product	$\log_b (MN) = \log_b M + \log_b N$	Log of a product is the sum of logs
Quotient	$\log_b (M/N) = \log_b M - \log_b N$	Log of a quotient is the difference of logs
Power	$\log_b (M^k) = k \log_b M$	Log of a power is the exponent times the log

Change of Base Property

The change of base property allows you to evaluate logarithms with any base using a calculator.

• Formula: $\log_b a = \frac{\log_c a}{\log_c b}$, where $c$ is any positive value (commonly 10 or $e$)

Solving Exponential Equations

Exponential equations can be solved by expressing both sides with the same base or by using logarithms.

• Same Base: If possible, rewrite both sides with the same base and set exponents equal.

• Using Logs: If not possible, take the logarithm of both sides and solve for the variable.

Solving Logarithmic Equations

Logarithmic equations can be solved by rewriting in exponential form or by using properties of logarithms to combine terms.

• Isolate the logarithmic expression if possible.

• Rewrite the equation in exponential form to solve for the variable.

• Check for extraneous solutions, especially if the argument of the log must be positive.