Difference between revisions of "1983 AIME Problems/Problem 1"
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With some substitution, we get <math>w^5w^3z^{120}=w^{10}</math> and <math>\log_zw=60</math>. | With some substitution, we get <math>w^5w^3z^{120}=w^{10}</math> and <math>\log_zw=60</math>. | ||
| − | + | == See also == | |
| − | + | {{AIME box|year=1983|before=First Question|num-a=2}} | |
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[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
Revision as of 17:34, 21 March 2007
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 the equivalent exponential expressions.
, 
, and 
. If we now convert everything to a power of 
, it will be easy to isolate and .
, 
, and 
.
With some substitution, we get 
and 
.
See also
| 1983 AIME (Problems • Answer Key • Resources) | ||
| Preceded by First Question |
Followed by Problem 2 | |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
| All AIME Problems and Solutions | ||
