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

Revision as of 14:14, 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.

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

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

Revision as of 14:14, 14 July 2018

Contents

Problem

Solution 1

Solution 2: Substitution

See also

@@ Line 25: / Line 25: @@
 \begin{align*}
 {\log_2 (\frac{1}{3}\log_2 x)} &= \frac{1}{3}{\log_2 (\log_2 x)}\\
-{\log_2 x = y}
+{\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)}\\