Exponential and Logarithmic Functions

These flashcards cover the essential vocabulary and concepts from the lecture on Exponential and Logarithmic Functions.

Logarithmic Function

An inverse function of an exponential function, denoted as log_a(x), that represents the exponent to which the base 'a' must be raised to obtain the number x.

Base of a Logarithm

The base 'a' in a logarithmic function log_a(x), which is a positive number not equal to 1.

Common Logarithm

A logarithm with base 10, denoted as log(x), which answers the question: to what power must 10 be raised to obtain x?

Natural Logarithm

A logarithm with base e, denoted as ln(x), where e is approximately equal to 2.71828.

One-to-One Function

A function in which each output value corresponds to exactly one input value, allowing it to have an inverse.

Exponential Form

The representation of a logarithm in its equivalent exponential form: if log_a(x) = y, then a^y = x.

Inverse Function Property

A property that states that if f(x) is a function, then f(f^{-1}(x)) = x, where f^{-1}(x) is the inverse function.

Domain of Logarithmic Functions

The set of all positive real numbers (0, ∞) for functions of the form f(x) = log_a(x).

Range of Logarithmic Functions

The set of all real numbers for functions of the form f(x) = log_a(x).

Vertical Asymptote

A vertical line x = c where the function approaches infinity as x approaches c from the left or right.

Graph of Logarithmic Function

The plot of the function f(x) = log_a(x), which typically starts at (1, 0), increases slowly, and has a vertical asymptote at x=0.

Logarithm Evaluating Example

For log_10(1000) = 3, since 10^3 = 1000, demonstrating how to evaluate logarithms by finding corresponding powers.

Horizontal Line Test

A test used to determine if a function is one-to-one: if any horizontal line crosses the graph more than once, the function is not one-to-one.

Reflecting Graphs of Logarithmic Functions

To obtain the graph of a logarithmic function, reflect the graph of its corresponding exponential function across the line y=x.