Difference between revisions of "2005 AMC 10B Problems/Problem 17"
Dairyqueenxd (talk | contribs) (→Solution) |
Dairyqueenxd (talk | contribs) (→Solution using logarithms) |
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\end{align*}</cmath> | \end{align*}</cmath> | ||
− | ==Solution | + | ==Solution 2 ([[logarithms]])== |
We can write <math>a</math> as <math>\log_4 5</math>, <math>b</math> as <math>\log_5 6</math>, <math>c</math> as <math>\log_6 7</math>, and <math>d</math> as <math>\log_7 8</math>. | We can write <math>a</math> as <math>\log_4 5</math>, <math>b</math> as <math>\log_5 6</math>, <math>c</math> as <math>\log_6 7</math>, and <math>d</math> as <math>\log_7 8</math>. | ||
− | |||
− | |||
− | < | + | We know that <math>\log_b a</math> can be rewritten as <math>\frac{\log a}{\log b}</math>, so we have: |
− | + | <cmath>\begin{align*} | |
− | <cmath>\frac{3\log2}{2\log2}= | + | a\cdot b \cdot c \cdot d &= \frac{\cancel{\log5}}{\log4}\cdot\frac{\cancel{\log6}}{\cancel{\log5}}\cdot\frac{\cancel{\log7}}{\cancel{\log6}}\cdot\frac{\log8}{\cancel{\log7}} \\ |
− | + | a\cdot b \cdot c \cdot d &= \frac{\log8}{\log4} \\ | |
− | + | a\cdot b \cdot c \cdot d &= \frac{3\cancel{\log2}}{2\cancel{\log2}} \\ | |
+ | a\cdot b \cdot c \cdot d &= \boxed{\textbf{(B) }\frac{3}{2}} \\ | ||
+ | \end{align*}</cmath> | ||
==Solution using logarithm chain rule== | ==Solution using logarithm chain rule== |
Revision as of 13:08, 16 December 2021
Contents
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 using logarithm chain rule
As in solution 2, we can write as
,
as
,
as
, and
as
.
is equivalent to
. Note that by the logarithm chain rule, this is equivalent to
, which evaluates to
, so
is the answer.
~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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