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.

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.

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