$y = \log_b(x)$

$\lim_{x \to 0^+} a\log_b(x) = \pm \infty$

$\lim_{x \to \infty} a\log_b(x) = \pm \infty$

$g(x) = \log_b(x + k)$

$b^y = x$

$\log_b(1) = 0$

$\log_b(b) = 1$

$\log_a(x) = \frac{\log_b(x)}{\log_b(a)}$

$\log_b(MN) = \log_b(M) + \log_b(N)$

$\log_b(\frac{M}{N}) = \log_b(M) - \log_b(N)$

The inverse of an exponential function; 'undoes' exponentiation.

x > 0 (positive real numbers)

All real numbers

A vertical line that the graph of a function approaches but never touches.

The value inside the logarithm, e.g., 'x' in $\log_b(x)$. Must be positive.

The value 'b' in $\log_b(x)$. Determines if the function is increasing or decreasing.

Describes the curve's shape: either concave up or concave down, but not both.

A transformation of a graph where the entire graph is moved left or right.

Describes how the function behaves as x approaches positive or negative infinity.

Changes to a function's graph, such as shifts, stretches, or reflections.

Logarithmic functions are the inverses of exponential functions. They 'undo' each other.

If b > 1, the function is increasing. If 0 < b < 1, the function is decreasing.

Because the inverse exponential function has a horizontal asymptote at y = 0, restricting the domain of the logarithm to positive values.

As x approaches infinity, y also approaches either positive or negative infinity, depending on the base and any transformations.

A horizontal shift changes the vertical asymptote and, consequently, the domain. For example, $\log_b(x+k)$ shifts the asymptote to x = -k, and the domain becomes x > -k.

Because they are always either increasing or decreasing and always concave up or concave down.

The argument of a logarithm must be positive because you cannot raise a base to any power and get a non-positive result.

It reflects the graph across the x-axis, changing increasing functions to decreasing and vice versa, and affecting the end behavior.

Vertical shifts and stretches can change the range but since the range is all real numbers, only vertical shifts change the graph.

If output values are proportional over equal input intervals, then the function is logarithmic.