Master Guide: AP Precalculus Unit 2 - Logarithmic Functions

Logarithm

An exponent or power that indicates the relationship between a base and an argument; the inverse function of exponential functions.

Inverse Function

A function that reverses another function; in logarithms, it reverses the action of exponentiation.

Base (b)

The number used as the base in a logarithmic equation, which must be greater than 0 and not equal to 1.

Argument (x)

The number for which the logarithm is being calculated, must be greater than 0.

Exponent (y)

The power to which the base must be raised to produce the argument.

Common Logarithm

A logarithm with base 10, denoted as log(x).

Natural Logarithm

A logarithm with base e, denoted as ln(x).

Product Property of Logarithms

States that the log of a product is the sum of the logs: logb(mn) = logb(m) + log_b(n).

Quotient Property of Logarithms

States that the log of a quotient is the difference of the logs: logb(m/n) = logb(m) - log_b(n).

Power Property of Logarithms

States that the exponent on the argument moves to the front: logb(m^p) = p * logb(m).

Change of Base Formula

Used to evaluate logarithms with bases other than those available on calculators: logb(a) = logc(a) / log_c(b).

Domain of Log Functions

The set of all positive real numbers, (0, ∞), for logarithmic functions.

Range of Log Functions

The set of all real numbers, (-∞, ∞), for logarithmic functions.

Vertical Asymptote

A line that a graph approaches but never touches or crosses, located at x = 0 for logarithmic functions.

Increasing Function

A function that rises as the input increases, applicable when base b > 1.

Decreasing Function

A function that falls as the input increases, applicable when 0 < b < 1.

Transformation of Log Functions

Changes to the graph of the logarithm based on vertical shifts, horizontal shifts, reflections, and dilations.

Vertical Shift

A transformation that moves a graph up or down, represented by 'k' in the function f(x) = a log_b(x - h) + k.

Horizontal Shift

A transformation that moves a graph left or right, represented by 'h' in the argument of the logarithm.

Reflection in the x-axis

A transformation that flips a graph over the x-axis, indicated by a negative coefficient in front of the logarithm.

Extraneous Solutions

Solutions that emerge from the algebraic process that do not satisfy the original logarithmic equation.

False Distributive Property

Mistake of assuming log(a + b) = log(a) + log(b), which is incorrect.

Logarithmic Equation

An equation that includes a logarithm, requiring specific methods to solve.

Solving Exponential Equations

The process of using logarithms to isolate and solve for the variable in an exponent.

Condensing Logarithms

The process of combining multiple logarithm expressions into a single logarithm.

Evaluating Logarithms

The process of determining the value of a logarithm based on its base and argument.