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.

A function of the form $f(x) = ab^x$, where a and b are constants, and b > 0 and not equal to 1.

A function of the form $f(x) = \log_b(x)$, which is the inverse of an exponential function.

A sequence where each term differs by a constant amount (common difference).

A sequence where each term is multiplied by a constant amount (common ratio).

A function formed by combining two or more functions, where the output of one function becomes the input of another.

Functions that 'undo' each other. If $f(g(x)) = x$ and $g(f(x)) = x$, then $f(x)$ and $g(x)$ are inverses.

A graph that uses a logarithmic scale on one axis and a linear scale on the other.

The constant amount added to each term to get the next term.

The constant amount multiplied by each term to get the next term.

The value of the function when x = 0 (the 'a' in $f(x) = ab^x$).

1. Isolate the exponential term. 2. Take the logarithm of both sides (using the same base as the exponential term). 3. Solve for the variable.

1. Isolate the logarithmic term. 2. Convert the equation to exponential form. 3. Solve for the variable. 4. Check for extraneous solutions.

1. Replace f(x) with y. 2. Swap x and y. 3. Solve for y. 4. Replace y with $f^{-1}(x)$.

1. Evaluate the inner function first. 2. Substitute the result into the outer function. 3. Simplify.

Show that $f(g(x)) = x$ and $g(f(x)) = x$.

1. Identify the initial value and growth/decay factor. 2. Plot key points. 3. Draw the curve, considering the asymptote.

1. Identify the base. 2. Find the vertical asymptote. 3. Plot key points. 4. Draw the curve, considering the asymptote.

Apply product rule, quotient rule, and power rule to combine or separate logarithmic terms.

1. Start with $f(x) = ab^x$. 2. Use two data points to create a system of equations. 3. Solve for a and b.

1. Set up the equation $0.5 = b^t$, where b is the decay factor and t is the half-life. 2. Solve for t using logarithms.