1983 AIME Problems/Problem 1
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
.
Alternative Solution
Applying Change of Base Formula:
Therefore,
.
Hence, .
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 |