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The given equation is:

\[
\log_{(x-4)}(3x) = 3
\]

To write this logarithmic equation in exponential form, we use the general rule for converting logarithmic equations to exponentials:

\[
\log_b(a) = c \quad \text{is equivalent to} \quad b^c = a
\]

In this case:
- \( b = x - 4 \) (the base of the logarithm),
- \( a = 3x \) (the argument inside the logarithm), and
- \( c = 3 \) (the right-hand side of the equation).

Thus, the exponential form of the equation is:

\[
(x - 4)^3 = 3x
\]

Let me know if you'd like further details or clarifications!

Here are 5 related questions:
1. How do you solve exponential equations like \( (x-4)^3 = 3x \)?
2. What is the significance of logarithmic and exponential relationships in real-life applications?
3. How can you determine the domain of a logarithmic function like \( \log_{(x-4)}(3x) \)?
4. Can you explain how to expand \( (x - 4)^3 \) using the binomial theorem?
5. How do logarithmic equations change if the base is a constant like 10 instead of a variable?

**Tip:** Remember, logarithms and exponentials are inverses of each other, which makes converting between the two quite straightforward!

Write the log equation as an exponential equation. You do not need to solve for x. log_{(x-4)}(3x) = 3

Learn how to convert the logarithmic equation log_{(x-4)}(3x) = 3 into its exponential form. This step-by-step guide explains key concepts of logarithms and exponents, including the formula log_b(a) = c is equivalent to b^c = a. Suitable for high school students (Grades 10-12).

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