2019 AIME II Problems/Problem 6
In a Martian civilization, all logarithms whose bases are not specified are assumed to be base , for some fixed . A Martian student writes down and finds that this system of equations has a single real number solution . Find .
Using change of base on the second equation to base b, Note by dolphin7 - you could also just rewrite the second equation in exponent form. Substituting this into the of the first equation,
We can manipulate this equation to be able to substitute a couple more times:
However, since we found that , is also equal to . Equating these,
We start by simplifying the first equation to Next, we simplify the second equation to Substituting this into the first equation gives Plugging this into gives -ktong
Apply change of base to to yield: which can be rearranged as: Apply log properties to to yield: Substituting into the equation yields: So Substituting this back in to yields So,
1st equation: 2nd equation: So now substitute and : We also have that This means that , so .
Solution 5 (Substitution)
Let Then we have which gives Plugging this in gives which gives so By substitution we have which gives Plugging in again we get
Solution 6 (Also Substitution)
This system of equations looks complicated to work with, so we let to make it easier for us to read.
Now, the first equation becomes .
The second equation, gives us .
Let's plug this back into the first equation to see what we get: , and simplifying, , so .
Combining this new finding with what we had above .
Now that we've expressed one variable in terms of the other, we can plug this into either equation, say equation 1. Then we get .
Finally, that gives us that . Thus, .
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