Transformations of Logarithmic Functions

Front

What is a logarithmic function?

Back

A logarithmic function is the inverse of an exponential function, typically expressed as f(x) = log_b(x), where b is the base.

Front

What is the base of a logarithm?

Back

The base of a logarithm is the number that is raised to a power to obtain a given number. For example, in log_b(x), b is the base.

Front

What does the graph of f(x) = log_b(x) look like?

Back

The graph of f(x) = log_b(x) is a curve that increases slowly, passing through the point (1,0) and approaching the vertical line x=0 (the y-axis) as an asymptote.

Front

What is the vertical asymptote of the logarithmic function f(x) = log_b(x)?

Back

The vertical asymptote of f(x) = log_b(x) is at x = 0.

Front

How does the graph of f(x) = log_b(x) change with a horizontal shift?

Back

A horizontal shift to the right by k units is represented as f(x) = log_b(x - k), while a shift to the left is represented as f(x) = log_b(x + k).

Front

What effect does a vertical shift have on the graph of a logarithmic function?

Back

A vertical shift upwards by k units is represented as f(x) = log_b(x) + k, while a shift downwards is represented as f(x) = log_b(x) - k.

Front

What does a reflection about the y-axis indicate in a logarithmic function?

Back

A reflection about the y-axis is represented as f(x) = log_b(-x), which flips the graph over the y-axis.

Front

What does a reflection about the x-axis indicate in a logarithmic function?

Back

A reflection about the x-axis is represented as f(x) = -log_b(x), which flips the graph over the x-axis.

Front

How do you identify the asymptote of the transformed function f(x) = log_b(-x + 1) - 5?

Back

The asymptote is found by setting the argument of the logarithm to zero: -x + 1 = 0, which gives x = 1.

Front

What is the relationship between logarithmic and exponential functions?

Back

Logarithmic functions are the inverse of exponential functions, meaning if y = b^x, then x = log_b(y).