1983 AIME Problems/Problem 1
Let , and all exceed and let be a positive number such that , and . Find .
The logarithmic notation doesn't tell us much, so we'll first convert everything to the equivalent exponential forms.
, , and . If we now convert everything to a power of , it will be easy to isolate and .
, , and .
With some substitution, we get and .
First we'll convert everything to exponential form. , , and . The only expression containing is . It now becomes clear that one way to find is to find what and are in terms of .
Taking the square root of the equation results in . Raising both sides of to the th power gives .
Going back to , we can substitute the and with and , respectively. We now have . Simplifying, we get . So our answer is .
Applying the change of base formula, Therefore, .
Since , the given conditions can be rewritten as , , and . Since , . Therefore, .
If we convert all of the equations into exponential form, we receive , , and . The last equation can also be written as . Also note that by multiplying the first two equations, we get, . Taking the square root of this, we find that . Recall, . Thus, . Also recall, . Therefore, = = . So, = .
Converting all of the logarithms to exponentials gives and Thus, we have We are looking for which by substitution, is
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