Logarithmic Functions and Applications (Video Notes)

Vocabulary flashcards covering the core terms and definitions from the lesson on logarithmic functions, including form conversions, properties, and graph features.

Logarithmic Function

A function of the form f(x) = log_b(x) with x > 0, b > 0, and b ≠ 1; it is the inverse of the exponential function and can be described by a table of values, an equation, or a graph.

Inverse of the Exponential Function

The logarithmic function; it undoes exponentiation, so if y = b^x then x = log_b(y).

Logarithmic Form

An equation written as log_b x = y, showing the exponent to which base b must be raised to obtain x.

Exponential Form

An equation written as x = b^y; the inverse relationship to logarithmic form.

Base (b)

The number in exponential and logarithmic expressions that is raised to a power; for logarithms, b > 0 and b ≠ 1.

Common Logarithm

Logarithm with base 10; written as log x.

Natural Logarithm

Logarithm with base e (Euler’s number); written as ln x.

Domain of Logarithmic Function

The set of x-values for which log_b(x) is defined; x > 0.

Range of Logarithmic Function

All real numbers (−∞, ∞) for log_b(x).

Intercept (Graph Intercepts)

Points where the graph crosses the axes; x-intercept occurs where y = 0; y-intercept occurs at a defined x value.

Zeros of Logarithmic Function

Points where the graph crosses the x-axis; for logb x, the zero is at x = 1 since logb(1) = 0.

Vertical Asymptote

A vertical line that the graph approaches but never touches; for log_b x this is x = 0; for shifted logs, the asymptote is where the inner argument equals zero.

Reflection Across y = x

The graphs of f(x) = log_b x and g(x) = b^x are inverses and reflect across the line y = x.

Log_b x = y implies x = b^y

Convert from logarithmic to exponential form.

x = b^y implies log_b x = y

Convert from exponential to logarithmic form.

Common Logarithm Notation

Logarithm with base 10, written as log x (no subscript).

Natural Logarithm Notation

Logarithm with base e, written as ln x.

Behavior by Base (b)

If b > 1, logb(x) is increasing; if 0 < b < 1, logb(x) is decreasing.

Argument of a Logarithm

The value inside the log, such as x in logb x or (x+c) in logb(x+c); determines domain restrictions and where vertical asymptotes occur.

Intercepts for Shifted Logarithms

For y = logb(x+c), the x-intercept and y-intercept depend on c; you solve logb(x+c) = 0 for x to find x-intercept and plug x into log_b(x+c) for y-intercept.

Zeros of Logarithmic Functions (Specific)

The x-value where f(x) = 0; for f(x) = log_b x this occurs at x = 1.

Table of Values for a Log Function

A set of ordered pairs (x, f(x)) used to describe or sketch f(x) = log_b x.

Sketching a Logarithmic Graph from Its Equation

Use the logarithmic form to derive key points via the exponential form and plot them to sketch the curve, noting the vertical asymptote and domain.