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Difference between revisions of "2003 AMC 12A Problems/Problem 24"

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== Problem ==
 
== Problem ==
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If <math>a\geq b > 1,</math> what is the largest possible value of <math>\log_{a}(a/b) + \log_{b}(b/a)?</math>
 
If <math>a\geq b > 1,</math> what is the largest possible value of <math>\log_{a}(a/b) + \log_{b}(b/a)?</math>
  
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</math>
 
</math>
  
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== Solution ==
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== Solution 1==
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Using logarithmic rules, we see that
 
Using logarithmic rules, we see that
  
 
<cmath>\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)</cmath>
 
<cmath>\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)</cmath>
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<cmath>=2-(\log_{a}b+\frac {1}{\log_{a}b})</cmath>
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<cmath>=2-\left(\log_{a}b+\frac {1}{\log_{a}b}\right)</cmath>
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Since <math>a</math> and <math>b</math> are both greater than <math>1</math>, using [[AM-GM]] gives that the term in parentheses must be at least <math>2</math>, so the largest possible values is <math>2-2=0 \Rightarrow \boxed{\textbf{B}}.</math>
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 +
Note that the maximum occurs when <math>a=b</math>.
 +
 
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==Solution 2 (Calculus)==
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By the logarithmic rules, we have <math>2-(\log_a{b}+\frac{1}{\log_a{b}})</math>.
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Let <math>x=\log_a{b}</math>. Thus, the expression becomes <math>2-(x+\frac{1}{x})</math>. We want to find the maximum of the function. To do so, we will find its derivative and let it equal to 0. We get: <math>\frac{d}{dx}\big(2-(x+\frac{1}{x})\big)=\frac{d}{dx}\big(2-x-\frac{1}{x})=-1+x^{-2}=0 \implies \frac{1}{x^2}=1, x^2=1, x=\pm1.</math>
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Since <math>a\geq b>1, x=-1</math> is not a solution. Thus, <math>x=1</math>. Substituting it into the original expression <math>2-(x+\frac{1}{x})</math>, we get <math>2-(1+\frac{1}{1})=2-2=\boxed{0}</math>.
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==Video Solution==
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The Link: https://www.youtube.com/watch?v=InF2phZZi2A&t=1s
  
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Since <math>a</math> and <math>b</math> are both positive, using [[AM-GM]] gives that the term in parentheses must be at least <math>2</math>, so the largest possible values is <math>2-2=\boxed{0}.</math>
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-MistyMathMusic
  
 
== See Also ==
 
== See Also ==
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{{AMC12 box|year=2003|ab=A|num-b=23|num-a=25}}
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{{AMC12 box|year=2003|ab=A|num-b=23|after=[[Problem 25]]}}
 
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:13, 20 June 2025

Problem

If $a\geq b > 1,$ what is the largest possible value of $\log_{a}(a/b) + \log_{b}(b/a)?$

$\mathrm{(A)}\ -2 \qquad \mathrm{(B)}\ 0 \qquad \mathrm{(C)}\ 2 \qquad \mathrm{(D)}\ 3 \qquad \mathrm{(E)}\ 4$

Solution 1

Using logarithmic rules, we see that

\[\log_{a}a-\log_{a}b+\log_{b}b-\log_{b}a = 2-(\log_{a}b+\log_{b}a)\] \[=2-\left(\log_{a}b+\frac {1}{\log_{a}b}\right)\]

Since $a$ and $b$ are both greater than $1$, using AM-GM gives that the term in parentheses must be at least $2$, so the largest possible values is $2-2=0 \Rightarrow \boxed{\textbf{B}}.$

Note that the maximum occurs when $a=b$.

Solution 2 (Calculus)

By the logarithmic rules, we have $2-(\log_a{b}+\frac{1}{\log_a{b}})$. Let $x=\log_a{b}$. Thus, the expression becomes $2-(x+\frac{1}{x})$. We want to find the maximum of the function. To do so, we will find its derivative and let it equal to 0. We get: $\frac{d}{dx}\big(2-(x+\frac{1}{x})\big)=\frac{d}{dx}\big(2-x-\frac{1}{x})=-1+x^{-2}=0 \implies \frac{1}{x^2}=1, x^2=1, x=\pm1.$ Since $a\geq b>1, x=-1$ is not a solution. Thus, $x=1$. Substituting it into the original expression $2-(x+\frac{1}{x})$, we get $2-(1+\frac{1}{1})=2-2=\boxed{0}$.

Video Solution

The Link: https://www.youtube.com/watch?v=InF2phZZi2A&t=1s

-MistyMathMusic

See Also

2003 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC Logo.png

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