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Difference between revisions of "2005 AMC 12B Problems/Problem 13"

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<math>\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</math>
 
<math>\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}</math>
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== Solution ==
 
== Solution ==
 
We see that we can re-write <math>4^{x_1}=5</math>, <math>5^{x_2}=6</math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math> as <math>\left(...\left(\left(\left(4^{x_1}\right)^{x_2}\right)^{x_3}\right)...\right)^{x_{124}}=128</math> by using substitution. By using the properties of exponents, we know that <math>4^{x_1x_2...x_{124}}=128</math>.  
 
We see that we can re-write <math>4^{x_1}=5</math>, <math>5^{x_2}=6</math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math> as <math>\left(...\left(\left(\left(4^{x_1}\right)^{x_2}\right)^{x_3}\right)...\right)^{x_{124}}=128</math> by using substitution. By using the properties of exponents, we know that <math>4^{x_1x_2...x_{124}}=128</math>.  
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==Alternate Solution==
 
==Alternate Solution==
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Changing <math>4^{x_1}=5</math> to logarithmic form, we get <math>{x_1}=\log_{4}{5}</math>.  We can rewrite this as <math>{x_1}=\dfrac{log_5}{log_4}</math>.  Applying this to the rest, we get <math>x_1x_2...x_{124}=\dfrac{log5}{log4}*\dfrac{log6}{log5}*...*\dfrac{log128}{log127}=\dfrac{log5*log6*...*log128}{log4*log5*...*log127}=\dfrac{log128}{log4}=log_4128=log_4{2^7}=7*log_42=7*\dfrac{1}{2}=\boxed{\mathrm{(D)}\,\dfrac{7}{2}}</math>
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Changing <math>4^{x_1}=5</math> to logarithmic form, we get <math>{x_1}=\log_{4}{5}</math>.  We can rewrite this as <math>{x_1}=\dfrac{\log{5}}{\log{4}}</math>.  Applying this to the rest, we get <math>x_1x_2...x_{124}=\dfrac{\log5}{\log4}\cdot\dfrac{\log6}{\log5}\cdot...\cdot\dfrac{\log128}{\log127}=\dfrac{\log5\cdot\log6\cdot...\cdot\log128}{\log4\cdot\log5\cdot...\cdot\log127}=\dfrac{\log128}{\log4}=\log_{4}{128}=\log_{4}{2^7}=7\cdot\log_{4}{2}=7\cdot\dfrac{1}{2}=\boxed{\mathrm{(D)}\,\dfrac{7}{2}}</math>
  
 
== See also ==
 
== See also ==
 
{{AMC12 box|year=2005|ab=B|num-b=12|num-a=14}}
 
{{AMC12 box|year=2005|ab=B|num-b=12|num-a=14}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 23:21, 7 March 2025

Problem

Suppose that $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$. What is $x_1x_2...x_{124}$?

$\mathrm{(A)}\ {{{2}}} \qquad \mathrm{(B)}\ {{{\frac{5}{2}}}} \qquad \mathrm{(C)}\ {{{3}}} \qquad \mathrm{(D)}\ {{{\frac{7}{2}}}} \qquad \mathrm{(E)}\ {{{4}}}$

Solution

We see that we can re-write $4^{x_1}=5$, $5^{x_2}=6$, $6^{x_3}=7$, ... , $127^{x_{124}}=128$ as $\left(...\left(\left(\left(4^{x_1}\right)^{x_2}\right)^{x_3}\right)...\right)^{x_{124}}=128$ by using substitution. By using the properties of exponents, we know that $4^{x_1x_2...x_{124}}=128$.

$4^{x_1x_2...x_{124}}=128\\2^{2x_1x_2...x_{124}}=2^7\\2x_1x_2...x_{124}=7\\x_1x_2...x_{124}=\dfrac{7}{2}$

Therefore, the answer is $\boxed{\mathrm{(D)}\,\dfrac{7}{2}}$

Alternate Solution

Changing $4^{x_1}=5$ to logarithmic form, we get ${x_1}=\log_{4}{5}$. We can rewrite this as ${x_1}=\dfrac{\log{5}}{\log{4}}$. Applying this to the rest, we get $x_1x_2...x_{124}=\dfrac{\log5}{\log4}\cdot\dfrac{\log6}{\log5}\cdot...\cdot\dfrac{\log128}{\log127}=\dfrac{\log5\cdot\log6\cdot...\cdot\log128}{\log4\cdot\log5\cdot...\cdot\log127}=\dfrac{\log128}{\log4}=\log_{4}{128}=\log_{4}{2^7}=7\cdot\log_{4}{2}=7\cdot\dfrac{1}{2}=\boxed{\mathrm{(D)}\,\dfrac{7}{2}}$

See also

2005 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 12 		Followed by
Problem 14
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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