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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.