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!