How many solutions does the equation [(a-b)log4(b/a) = -6] have if (a-b) is an integer?

Question

How many solutions does the equation $$ (a-b)\log_4(\frac{b}{a}) = -6 $$ have if $(a-b)$ is an integer?

My Solution:

I assume b-a = n, an integer. Substituting values and simplifying I get

$$1 + \frac{n}{a} = 2^{\frac{12}{n}} $$

Now I can substitute any value of n and get the value for a and b. Which means this equation has infinite solutions. However, the book I took this question from gives the answer 12. The book does not provide any solution.

Answer 1

I think $a$ has to be rational.

Hint to continue:

From $1+\frac{n}{a}=2^\frac{12}{n}$, we can see that both sides of the equation are rational. As such, it can be shown that $\frac{12}{n}$ has to be an integer. There are $12$ factors (positive and negative) of $12$ which $n$ can be.