Difference between revisions of "1988 AIME Problems/Problem 3"
(→Solution 3) |
(→Solution 3) |
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Line 60: | Line 60: | ||
\end{align*}</cmath> | \end{align*}</cmath> | ||
Now setting <math>y=\log_2x</math>, we have | Now setting <math>y=\log_2x</math>, we have | ||
− | <cmath>\sqrt[3]y=\frac{y}{3}</cmath> | + | <cmath>\sqrt[3]{y}=\frac{y}{3}</cmath> |
− | Solving gets | + | Solving gets <math>y=\log_2x=3\sqrt{3}\Longrightarrow(\log_2x)^2=(3\sqrt3)^2=\boxed{27}</math>. |
== See also == | == See also == |
Revision as of 19:04, 26 August 2019
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
.
See also
1988 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |
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