(11) PC I Chapter 3.2: Logarithmic Functions

Logarithm

A function that determines the power to which a base must be raised to produce a given number.

Inverse Functions

Two functions that reverse the effect of each other; for logarithmic and exponential functions, they switch their input and output.

Base of a Logarithm

The number that is raised to a power; in logarithm notation, it is the subscript of log (e.g., in log_b(x), b is the base).

Domain of a Logarithmic Function

The set of input values for which the logarithmic function is defined; for log_b(x), it is (0, ∞).

Range of a Logarithmic Function

The set of output values of a logarithmic function; for log_b(x), it is (-∞, ∞).

Common Logarithm

A logarithm with base 10, often written as log(x) without explicitly noting the base.

Natural Logarithm

A logarithm with base e, denoted as ln(x), where e is approximately 2.718.

Logarithmic Conversion

The method of rewriting logarithmic expressions in exponential form; for example, log_b(x) = y becomes b^y = x.

Vertical Asymptote

A vertical line that the graph approaches but never touches; for logarithmic functions, it is found at x = 0.

Horizontal Asymptote

A horizontal line that the graph approaches as x approaches infinity; for exponential functions, it is y = 0.

Product Property of Logarithms

The logarithm of a product is the sum of the logarithms of the factors: log_b(m*n) = log_b(m) + log_b(n).

Quotient Property of Logarithms

The logarithm of a quotient is the difference of the logarithms: log_b(m/n) = log_b(m) - log_b(n).

Power Property of Logarithms

The logarithm of a number raised to a power is the exponent times the logarithm of the base: log_b(m^p) = p*log_b(m).

Logarithmic Shift

A transformation applied to a logarithmic function that changes its graph; horizontal shifts affect the domain and vertical shifts affect the range.

Reflection Over x-axis

A transformation in which the graph of a function is flipped over the x-axis, affecting its output values.

Reflection Over y-axis

A transformation in which the graph of a function is flipped over the y-axis, affecting its input values.