Glossary

Additive Transformations

Criticality: 3

Changes to a function's graph that involve adding or subtracting constants, resulting in shifts (translations) of the graph.

Example:

Shifting the graph of $y = \log_2(x)$ to the left by 3 units to get $y = \log_2(x+3)$ is an Additive Transformation.

Argument of a Logarithm

Criticality: 3

The expression inside the logarithm, which must always be a positive value for the logarithm to be defined in the real number system.

Example:

In $\log_7(2x-5)$ , the Argument of a Logarithm is $(2x-5)$ , which must be greater than zero for the function to exist.

Base of the Logarithm

Criticality: 3

The number that is raised to a power in an exponential expression, and which determines the growth or decay rate and direction of a logarithmic function.

Example:

In $\log_3(x)$ , the Base of the Logarithm is 3, which indicates the function is increasing.

Concavity (Logarithmic Functions)

Criticality: 2

Describes the direction in which the graph of a function opens, either upward (concave up) or downward (concave down).

Example:

The graph of $y = \log_2(x)$ exhibits Concavity that is always downward, curving like a frown.

Domain (of Logarithmic Functions)

Criticality: 3

The set of all possible input values (x-values) for which a logarithmic function is defined, which must always be positive numbers.

Example:

For $f(x) = \log_2(x-3)$ , the Domain is $x > 3$ because the argument $(x-3)$ must be greater than zero.

End Behavior (Logarithmic Functions)

Criticality: 3

Describes the trend of the function's output values as the input values approach positive or negative infinity, or specific critical points.

Example:

The End Behavior of $y = \log_2(x)$ shows that as x approaches 0 from the right, y approaches negative infinity, plummeting downwards.

Extrema (Logarithmic Functions)

Criticality: 1

The maximum or minimum values of a function. Logarithmic functions typically do not have global extrema.

Example:

Because a basic Logarithmic Function continuously increases or decreases, it does not have any Extrema like a peak or a valley.

Horizontal Shifts

Criticality: 3

A type of transformation that moves a graph left or right by adding or subtracting a constant inside the function's argument.

Example:

The function $g(x) = \log_b(x + k)$ demonstrates a Horizontal Shift of the graph of $f(x) = \log_b(x)$ by $k$ units.

Increasing/Decreasing (Logarithmic Functions)

Criticality: 2

Describes whether the function's output values are consistently rising or falling as the input values increase.

Example:

A Logarithmic Function with a base greater than 1, like $y = \log_5(x)$ , is always Increasing as x gets larger.

Inflection Point (Logarithmic Functions)

Criticality: 1

A point on a curve where the concavity changes. Logarithmic functions typically do not have inflection points.

Example:

Unlike a cubic function, a standard Logarithmic Function maintains the same concavity throughout its domain, thus having no Inflection Points.

Inverse Function

Criticality: 2

A function that 'undoes' the action of another function, meaning if f(a)=b, then f⁻¹(b)=a.

Example:

Addition and subtraction are Inverse Functions; if you add 5, subtracting 5 perfectly reverses the operation.

Limits (of Logarithmic Functions)

Criticality: 2

A mathematical concept describing the value that a function approaches as the input approaches some value.

Example:

The Limit $\lim_{x \to 0^+} \log_b(x) = -\infty$ (for $b>1$ ) formally describes the function's behavior near its vertical asymptote.

Logarithmic Function

Criticality: 3

The inverse of an exponential function, used to find the exponent to which a base must be raised to produce a given number.

Example:

If you want to find out what power of 10 gives you 1000, you'd use a Logarithmic Function: log₁₀(1000) = 3.

Range (of Logarithmic Functions)

Criticality: 2

The set of all possible output values (y-values) that a logarithmic function can produce, which includes all real numbers.

Example:

No matter how large or small the positive input, the Range of $y = \log_b(x)$ will cover every real number on the y-axis, extending infinitely up and down.

Vertical Asymptote (Logarithmic Functions)

Criticality: 3

A vertical line that the graph of a function approaches but never touches, indicating where the function is undefined.

Example:

For the function $y = \log_5(x)$ , the y-axis (or $x=0$ ) acts as a Vertical Asymptote, a boundary the graph gets infinitely close to.