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Difference between revisions of "1988 AIME Problems/Problem 3"

(→‎Solution 2: Substitution)
(→‎Solution 2: Substitution)
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{\log_2 (\frac{1}{3}y)} &= \frac{1}{3}{\log_2 (y)}\\
 
{\log_2 (\frac{1}{3}y)} &= \frac{1}{3}{\log_2 (y)}\\
 
{3\log_2 (\frac{1}{3}y)} &= {\log_2 (y)}\\  
 
{3\log_2 (\frac{1}{3}y)} &= {\log_2 (y)}\\  
 +
{\log_2 (\frac{1}{3}y)^3} &= {\log_2 (y)}\\
 
\end{align*}
 
\end{align*}
 
</cmath>
 
</cmath>
Line 35: Line 36:
  
 
----
 
----
 +
Just a quick note-
 +
In this solution, we used 2 important rules of logarithm:
 +
a) <math>\log_a b^n=n\log_a b</math>
 +
b) <math>\log_{a^n} b=\frac{1}{n}\log_a b</math>
  
 
== See also ==
 
== See also ==

Revision as of 14:16, 14 July 2018

Problem

Find $(\log_2 x)^2$ if $\log_2 (\log_8 x) = \log_8 (\log_2 x)$.

Solution 1

Raise both as exponents with base 8:

\begin{align*} 8^{\log_2 (\log_8 x)} &= 8^{\log_8 (\log_2 x)}\\ 2^{3 \log_2(\log_8x)} &= \log_2x\\ (\log_8x)^3 &= \log_2x\\ \left(\frac{\log_2x}{\log_28}\right)^3 &= \log_2x\\ (\log_2x)^2 &= (\log_28)^3 = \boxed{27}\\ \end{align*}

A quick explanation of the steps: On the 1st step, we use the property of logarithms that $a^{\log_a x} = x$. On the 2nd step, we use the fact that $k \log_a x = \log_a x^k$. On the 3rd step, we use the change of base formula, which states $\log_a b = \frac{\log_k b}{\log_k a}$ for arbitrary $k$.

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.

\begin{align*} {\log_2 (\frac{1}{3}\log_2 x)} &= \frac{1}{3}{\log_2 (\log_2 x)}\\ {\log_2 x = y}\\ {\log_2 (\frac{1}{3}y)} &= \frac{1}{3}{\log_2 (y)}\\ {3\log_2 (\frac{1}{3}y)} &= {\log_2 (y)}\\ {\log_2 (\frac{1}{3}y)^3} &= {\log_2 (y)}\\ \end{align*} Solving, we get $y^2 = 27$, which is what we want. $\boxed{27}$


Just a quick note- In this solution, we used 2 important rules of logarithm: a) $\log_a b^n=n\log_a b$ b) $\log_{a^n} b=\frac{1}{n}\log_a b$

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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