The given problem is to solve for $x$ in the equation:

$\log_x(27) = 3$

Step-by-Step Solution:

1. Recall the logarithmic definition: $\log_a(b) = c \quad \text{means} \quad a^c = b$

   Here, $a = x$, $b = 27$, and $c = 3$. Using this definition: $x^3 = 27$

2. Solve for $x$: $x = \sqrt[3]{27}$

   The cube root of $27$ is $3$, so: $x = 3$

Final Answer:

$x = 3$

Let me know if you'd like me to explain this further or if you have any related questions!

Related Questions:

1. How do logarithmic functions relate to exponential functions?
2. What are the properties of logarithms, such as the change of base formula?
3. How do you solve logarithmic equations when the base isn't clear?
4. How can logarithms be used to simplify complex exponential problems?
5. What are real-world applications of logarithms?

Tip:

Always remember that logarithms are the inverse of exponents, so converting between them is the key to solving problems efficiently!