2003 AIME I Problems/Problem 4
Given that and that find
Using the properties of logarithms, we can simplify the first equation to . Therefore,
Now, manipulate the second equation.
By the Pythagorean identities, , and we can substitute the value for from . .
Examining the first equation, we simplify as the following:
With this in mind, examining the second equation, we may simplify as the following (utilizing logarithm properties):
From here, we may divide both sides by and then proceed with the change-of-base logarithm property:
Thus, exponentiating both sides results in . Squaring both sides gives us
Via the Pythagorean Identity, and is simply , via substitution. Thus, substituting these results into the current equation:
Using simple cross-multiplication techniques, we have , and thus . ~ nikenissan
By the first equation, we get that . We can let , . Thus . By the identity , we get that . Solving this, we get . So we have
From here it is obvious that .
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