Difference between revisions of "1983 AIME Problems/Problem 1"
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== Solution == | == Solution == | ||
+ | The logarithmic notation doesn't tell us much, so we'll first convert everything to exponents. | ||
+ | |||
+ | <math>x^{24}=w</math>, <math>y^{40}=w</math>, and <math>(xyz)^{12}=w</math>. If we now convert everything to a power of <math>120</math>, it will be easy to isolate <math>z</math> and <math>w</math>. | ||
+ | |||
+ | <math>x^{120}=w^5</math>, <math>y^{120}=w^3</math>, and <math>(xyz)^{120}=w^{10}</math>. | ||
+ | |||
+ | With some substitution, we get <math>w^5w^3z^{120}=w^{10}</math> and <math>\log_zw=60</math>. | ||
---- | ---- |
Revision as of 23:49, 23 July 2006
Problem
Let ,
, and
all exceed
, and let
be a positive number such that
,
, and
. Find
.
Solution
The logarithmic notation doesn't tell us much, so we'll first convert everything to exponents.
,
, and
. If we now convert everything to a power of
, it will be easy to isolate
and
.
,
, and
.
With some substitution, we get and
.