The given equation is:

$2 \log_3 X - \log_3 4X = 2$

Step 1: Use logarithmic properties

Recall the following logarithmic rules:

1. $a \log_b M = \log_b M^a$
2. $\log_b M - \log_b N = \log_b \left(\frac{M}{N}\right)$

Apply these rules to simplify the left-hand side: $2 \log_3 X = \log_3 X^2$ $\log_3 X^2 - \log_3 4X = \log_3 \left(\frac{X^2}{4X}\right)$

Simplify the fraction: $\frac{X^2}{4X} = \frac{X}{4}$

Thus, the equation becomes: $\log_3 \left(\frac{X}{4}\right) = 2$

Step 2: Exponentiate to remove the logarithm

If $\log_b M = N$, then $M = b^N$. Apply this to the equation: $\frac{X}{4} = 3^2$

Simplify: $\frac{X}{4} = 9$

Multiply through by 4: $X = 36$

Final Answer:

$X = 36$

Would you like me to go over the properties of logarithms used here or any specific steps?

Questions for Practice:

1. How do you solve logarithmic equations involving addition instead of subtraction?
2. Can you verify if $X = 36$ satisfies the original equation?
3. What happens if the base of the logarithm is different (e.g., $\log_5$)?
4. How do you solve $\log_a (M) = \log_a (N)$?
5. How can logarithmic equations be graphed?

Tip:

Always check the domain of the logarithmic function to ensure that the solution is valid (logarithms are undefined for non-positive arguments).