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 precalculus, where the variable appears in the exponent. These functions model rapid growth or decay and are essential in many scientific and mathematical contexts.

• 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.

• 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.

The Number $e$

The number $e$ is a special irrational constant approximately equal to 2.71828. It is the base of the natural exponential function and arises in many contexts involving continuous growth or decay.

• Definition: $e \approx 2.718281828459$

• Exponential Function with Base $e$: $F(x) = e^x$

• Applications: Compound interest, population growth, radioactive decay, and more.

Example: Evaluate $e^2$ using a calculator.

Graphing Exponential Functions

Exponential functions have distinctive graphs that show rapid increase (for $a > 1$) or rapid decrease (for $0 < a < 1$). Their domain is all real numbers, and their range is always positive real numbers.

• Key Features:

  • Domain: $(-\infty, \infty)$

  • Range: $(0, \infty)$

  • Horizontal asymptote: $y = 0$

• Growth vs. Decay:

  • If $a > 1$, the function increases as $x$ increases (growth).

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

Transformations of Exponential Graphs

Exponential graphs can be shifted, reflected, stretched, or compressed using transformations. The general form is $g(x) = a \cdot f(b(x - h)) + k$.

• Vertical Shifts: $k$ moves the graph up or down.

• Horizontal Shifts: $h$ moves the graph left or right.

• Reflections: Negative signs reflect the graph across the x- or y-axis.

• Stretches/Compressions: Multiplying by $a$ or $b$ stretches or compresses the graph.

Example: Graph $g(x) = 2^{x-1} + 3$ by shifting $f(x) = 2^x$ right by 1 and up by 3.

Logarithmic Functions

Introduction to Logarithms

Logarithms are the inverses of exponential functions. The logarithm base $a$ of $x$ answers the question: "To what power must $a$ be raised to get $x$?"

• Definition: $y = \log_a x$ if and only if $a^y = x$

• Common Logarithms: Base 10 ($\log x$)

• Natural Logarithms: Base $e$ ($\ln x$)

Exponential Form	Logarithmic Form
$a^y = x$	$y = \log_a x$

Example: $2^3 = 8$ is equivalent to $\log_2 8 = 3$.

Evaluating Logarithms and Properties

Logarithms can be evaluated using their properties, which simplify expressions and solve equations.

• Product Rule: $\log_a (xy) = \log_a x + \log_a y$

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

• Power Rule: $\log_a (x^k) = k \log_a x$

• Change of Base: $\log_a x = \frac{\log_b x}{\log_b a}$

Name	Example	Property	Description
Product	$\log_a (xy)$	$\log_a x + \log_a y$	Log of a product is the sum
Quotient	$\log_a \left(\frac{x}{y}\right)$	$\log_a x - \log_a y$	Log of a quotient is the difference
Power	$\log_a (x^k)$	$k \log_a x$	Log of a power is the exponent times the log

Example: $\log_2 8 = 3$ because $2^3 = 8$.

Graphing Logarithmic Functions

Logarithmic functions are the inverses of exponential functions. Their graphs reflect exponential graphs over the line $y = x$.

• Key Features:

  • Domain: $(0, \infty)$

  • Range: $(-\infty, \infty)$

  • Vertical asymptote: $x = 0$

• Growth/Decay: For $a > 1$, $\log_a x$ increases; for $0 < a < 1$, $\log_a x$ decreases.

Transformations of Logarithmic Graphs

Logarithmic graphs can be shifted, reflected, stretched, or compressed using transformations, similar to exponential functions.

• General Form: $g(x) = a \cdot \log_b (x - h) + k$

• Vertical Shifts: $k$ moves the graph up or down.

• Horizontal Shifts: $h$ moves the graph left or right.

• Reflections: Negative signs reflect the graph across the x- or y-axis.

Example: Graph $g(x) = \log_2 (x - 1) + 3$ by shifting $f(x) = \log_2 x$ right by 1 and up by 3.

Properties of Logarithms: Product, Quotient, and Power Rules

These properties allow us to expand or condense logarithmic expressions, making them easier to evaluate or solve.

• Product Rule: $\log_a (xy) = \log_a x + \log_a y$

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

• Power Rule: $\log_a (x^k) = k \log_a x$

Property	Example	Description
Product	$\log_2 (8 \cdot 4) = \log_2 8 + \log_2 4$	Sum of logs
Quotient	$\log_2 \left(\frac{8}{4}\right) = \log_2 8 - \log_2 4$	Difference of logs
Power	$\log_2 (8^2) = 2 \log_2 8$	Exponent times log

Evaluating Logarithms: Change of Base Property

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

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

• Application: Use this property to compute logarithms not available on your calculator.

Example: $\log_2 11 = \frac{\log_{10} 11}{\log_{10} 2}$

Solving Exponential and Logarithmic Equations

Solving Exponential Equations Using Like Bases

To solve exponential equations, rewrite both sides with the same base if possible, then set the exponents equal to each other.

• Example: $2^x = 8$ can be rewritten as $2^x = 2^3$, so $x = 3$.

• Common Powers: Know the powers of small integers for quick rewriting.

Solving Exponential Equations Using Logarithms

If you cannot rewrite both sides with the same base, use logarithms to solve for the variable.

• Take the logarithm of both sides: $a^x = b$ becomes $x = \log_a b$ or $x = \frac{\log b}{\log a}$

• Apply properties of logarithms as needed.

Example: Solve $3^x = 7$ by taking $\log$ of both sides: $x = \frac{\log 7}{\log 3}$

Solving Logarithmic Equations

To solve logarithmic equations, use properties of logarithms to combine or expand expressions, then rewrite in exponential form if necessary.

• Example: $\log_2 (x + 6) = 3$ is equivalent to $x + 6 = 2^3 = 8$, so $x = 2$.

• Check for extraneous solutions: The argument of a logarithm must be positive.