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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?

equation

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:

equation

equation

Therefore, the domain is:

equation

Use the fact that:

equation

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

Make the change of variable:

equation

and the equation becomes:

equation

equation

a quadratic equation with roots:

equation

Now compute the solutions:

equation

equation

The first equation leads to:

equation

equation

equation

and the second equation:

equation

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

equation

with the roots:

equation

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