1988 AIME Problems/Problem 3
Problem
Find if .
Solution 1
Raise both as exponents with base 8:
A quick explanation of the steps: On the 1st step, we use the property of logarithms that . On the 2nd step, we use the fact that . On the 3rd step, we use the change of base formula, which states for arbitrary .
Solution 2: Substitution
We wish to convert this expression into one which has a uniform base. Let's scale down all the powers of 8 to 2.
Solving, we get , which is what we want.
Just a quick note- In this solution, we used 2 important rules of logarithm: 1) . 2) .
Solution 3
First we have Changing the base in the numerator yields Using the property yields Now setting , we have Solving gets .
~ Nafer
Solution 4
Say that and so we have . And we want .
Because (as and from our setup), we have that
~thedodecagon
Note that we use the property in step 1 and in step 2 in this solution.
See also
1988 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
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