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This problem trains for: AMC-10, AMC-12, AIME, SAT-II, Math Kangaroo 9-10.

How many solutions does the equation below have?

As a first step, establish the domain for the variable x. Any solutions found must be elements of this domain.

The conditions for the logarithms to exist are:

Therefore, the domain is:

Use the fact that:

which is a special case of the change of base formula.

Make the change of variable:

and the equation becomes:

a quadratic equation with roots:

Now compute the solutions:

The first equation leads to:

and the second equation:

clearly has a root equal to 1. We find that it can be factored as:

with the roots:

None of these roots are in the domain and, therefore, the equation has no solutions.