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Difference between revisions of "2002 AMC 12P Problems/Problem 21"

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== Solution ==
 
== Solution ==
 
We may rewrite the given equation as <math>2(\frac {\log c}{\log a} + \frac {\log c}{\log b}) = \frac {9\log c}{\log a + \log b}</math>.
 
We may rewrite the given equation as <math>2(\frac {\log c}{\log a} + \frac {\log c}{\log b}) = \frac {9\log c}{\log a + \log b}</math>.
−
Since <math>c \neq 1</math>, we have <math>\log c \neq 0</math>, so we may divide by <math>\log c</math> on both sides. After making the substitutions <math>x = \log a</math> and <math>y = \log b</math>, our equation becomes
+
Since <math>c \neq 1</math>, we have <math>\log c \neq 0</math>, so we may divide by <math>\log c</math> on both sides. After making the substitutions <math>x = \log a</math> and <math>y = \log b</math>, our equation becomes <cmath>\frac {2}{x} + \frac {2}{y} = \frac {9}{x+y}</cmath>
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2002|ab=P|num-b=20|num-a=22}}
 
{{AMC12 box|year=2002|ab=P|num-b=20|num-a=22}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 14:54, 10 March 2024

Problem

Let $a$ and $b$ be real numbers greater than $1$ for which there exists a positive real number $c,$ different from $1$, such that

\[2(\log_a{c} + \log_b{c}) = 9\log_{ab}{c}.\]

Find the largest possible value of $\log_a b.$

$\text{(A) }\sqrt{2} \qquad \text{(B) }\sqrt{3} \qquad \text{(C) }2 \qquad \text{(D) }\sqrt{6} \qquad \text{(E) }3$

Solution

We may rewrite the given equation as $2(\frac {\log c}{\log a} + \frac {\log c}{\log b}) = \frac {9\log c}{\log a + \log b}$. Since $c \neq 1$, we have $\log c \neq 0$, so we may divide by $\log c$ on both sides. After making the substitutions $x = \log a$ and $y = \log b$, our equation becomes \[\frac {2}{x} + \frac {2}{y} = \frac {9}{x+y}\]

See also

2002 AMC 12P (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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