Difference between revisions of "2005 AMC 10B Problems/Problem 17"
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\end{align*}</cmath> | \end{align*}</cmath> | ||
− | ==Solution | + | ==Solution 3 (logarithm chain rule)== |
− | As in | + | As in Solution 2, we can write <math>a</math> as <math>\log_4 5</math>, <math>b</math> as <math>\log_56</math>, <math>c</math> as <math>\log_67</math>, and <math>d</math> as <math>\log_78</math>. <math>a\cdot b\cdot c\cdot d</math> is equivalent to <math>(\log_4 5)\cdot (\log_5 6)\cdot (\log_6 7)\cdot (\log_7 8)</math>. By the logarithm chain rule, this is equivalent to <math>\log_4 8</math>, which evaluates to <math>\boxed{\textbf{(B) }\frac{3}{2}}</math>. |
+ | |||
~solver1104 | ~solver1104 | ||
Latest revision as of 14:30, 16 December 2021
Contents
[hide]Problem
Suppose that , , , and . What is ?
Solution
Solution 2 (logarithms)
We can write as , as , as , and as .
We know that can be rewritten as , so we have:
Solution 3 (logarithm chain rule)
As in Solution 2, we can write as , as , as , and as . is equivalent to . By the logarithm chain rule, this is equivalent to , which evaluates to .
~solver1104
See Also
2005 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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 10 Problems and Solutions |
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