The problem asks for the values of $a$ and $b$ in the exponential function $f(x) = ab^x$, which passes through the points $(0, 3)$ and $(2, 27)$.

Step-by-step solution:

1. Substitute the first point $(0, 3)$: When $x = 0$, $f(0) = ab^0 = a \cdot 1 = a$.
   So, $a = 3$.

2. Substitute the second point $(2, 27)$: When $x = 2$, $f(2) = ab^2$. Using $a = 3$: $27 = 3b^2.$

3. Solve for $b^2$: $b^2 = \frac{27}{3} = 9.$

4. Solve for $b$: Since $b$ is positive in exponential functions: $b = \sqrt{9} = 3.$

Final answer:

$a = 3, \quad b = 3.$

Would you like additional details or further clarification? Here are 5 related questions you might explore:

1. How would the solution change if the function had a negative $b$?
2. What are the implications of $a$ and $b$ values on the graph of $f(x) = ab^x$?
3. Can this method be used for other types of exponential functions, such as those with base $e$?
4. How does changing $b$ affect the growth or decay of the exponential function?
5. What happens if the two points provided are vertically aligned (same $x$-coordinate)?

Tip: Always start solving exponential functions by isolating $a$ using the point where $x = 0$, as this simplifies the calculations.