Difference between revisions of "2005 AMC 12B Problems/Problem 13"
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== Problem == | == Problem == | ||
+ | Suppose that <math>4^{x_1}=5</math>, <math>5^{x_2}=6</math>, <math>6^{x_3}=7</math>, ... , <math>127^{x_{124}}=128</math>. What is <math>x_1x_2...x_{124}</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> | ||
== 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>. | ||
+ | |||
+ | <math>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}</math> | ||
+ | |||
+ | Therefore, the answer is <math>\boxed{\mathrm{(D)}\,\dfrac{7}{2}}</math> | ||
+ | |||
+ | ==Alternate Solution== | ||
+ | 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}} | |
+ | {{MAA Notice}} |
Latest revision as of 18:57, 24 December 2020
Problem
Suppose that , , , ... , . What is ?
Solution
We see that we can re-write , , , ... , as by using substitution. By using the properties of exponents, we know that .
Therefore, the answer is
Alternate Solution
Changing to logarithmic form, we get . We can rewrite this as . Applying this to the rest, we get
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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