Arithmetic: Constant difference, linear growth | Geometric: Constant ratio, exponential growth

Growth: b > 1, increasing function | Decay: 0 < b < 1, decreasing function

Exponential: Rapid growth/decay, horizontal asymptote | Logarithmic: Slow growth, vertical asymptote

$f^{-1}(x)$: Inverse function | $\frac{1}{f(x)}$: Reciprocal of the function

Linear: Equal intervals represent equal changes | Logarithmic: Equal intervals represent equal proportional changes

Composite: Combining functions sequentially | Inverse: Undoing a function

Exponential: Domain: all reals, Range: positive reals | Logarithmic: Domain: positive reals, Range: all reals

Exponential: Isolate, take logarithm | Logarithmic: Isolate, convert to exponential

Exponential: Horizontal asymptote, rapid growth/decay | Logarithmic: Vertical asymptote, slow growth

Exponential: Larger base = faster growth/decay | Logarithmic: Base influences steepness and position

Exponential growth occurs when a quantity increases proportionally to its current value, leading to rapid growth over time.

Exponential decay occurs when a quantity decreases proportionally to its current value, leading to rapid decline over time.

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

If b > 1, the function represents growth. If 0 < b < 1, the function represents decay.

The base determines the rate at which the logarithm increases; it also defines the domain (x > 0).

Exponential functions have a horizontal asymptote, while logarithmic functions have a vertical asymptote. These are lines the graph approaches but never touches.

The domain is all real numbers, and the range is all positive real numbers (if a > 0).

The domain is all positive real numbers, and the range is all real numbers.

The inner function's output becomes the input for the outer function, applying transformations sequentially.

Graphically, a function and its inverse are reflections of each other across the line y=x.

$u_n = a + (n-1)d$, where 'a' is the first term and 'd' is the common difference.

$u_n = ar^{n-1}$, where 'a' is the first term and 'r' is the common ratio.

$f(x) = ab^x$

$f^{-1}(x)$

$f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$

$(f \circ g)(x) = f(g(x))$

Swap x and y, then solve for y.

$y = \log_b(x)$

$b^y = x$

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