What are the restrictions on the variable in the equation log(3x-5)-log(x-2)=log(x^2-5)?

5/3, 2, ???

To determine the restrictions on the variable in the given equation, we need to consider the domain of the logarithmic function. The logarithm function is defined only for positive values, so the argument of the logarithm must be greater than zero.

In the given equation, we have log(3x-5) - log(x-2) = log(x^2-5).

First, let's consider log(3x-5). The argument 3x-5 should be greater than zero, so we have:

3x - 5 > 0

Solving this inequality, we get:

3x > 5
x > 5/3

Next, let's consider log(x-2). The argument x-2 should be greater than zero, so we have:

x - 2 > 0

Solving this inequality, we get:

x > 2

Lastly, let's consider log(x^2-5). The argument x^2-5 should be greater than zero, so we have:

x^2 - 5 > 0

Solving this inequality, we get:

x^2 > 5
x > sqrt(5) ≈ 2.236

Therefore, the restrictions on the variable x are:

x > 5/3 and x > 2 and x > sqrt(5)

Since x must satisfy all three conditions, the smallest value that satisfies all the conditions is x = sqrt(5) ≈ 2.236.