1984 AIME Problems/Problem 5
Contents
Problem
Determine the value of if
and
.
Solutions
Solution 1
Use the change of base formula to see that ; combine denominators to find that
. Doing the same thing with the second equation yields that
. This means that
and that
. If we multiply the two equations together, we get that
, so taking the fourth root of that,
.
Solution 2
We can simplify our expressions by changing everything to a common base and by pulling exponents out of the logarithms. The given equations then become and
. Adding the equations and factoring, we get
. Rearranging we see that
. Again, we pull exponents out of our logarithms to get
. This means that
. The left-hand side can be interpreted as a base-2 logarithm, giving us
.
Solution 3
This solution is very similar to the above two, but it utilizes the well-known fact that Thus,
Similarly,
Adding these two equations, we have
.
Solution 4
We can change everything to a common base, like so:
We set the value of
to
, and the value of
to
Now we have a system of linear equations:
We multiply the second equation by three and subtract the first equation from the second to get
which gives
We substitute our value into the first equation and find that
Now we know that
and
so we find that
and
Multiplying together gives us
Solution 5
Add the two equations to get . This can be simplified with the log property
. Using this, we get
. Now let
and
. Converting to exponents, we get
and
. Sub in the
to get
. So now we have that
and
which gives
,
. This means
so
Solution 6
Add the equations and use the facts that and
to get
Now use the change of base identity with base as 2:
Which gives:
Solving gives,
Solution 7
By properties of logarithms, we know that .
Using the fact that , we get
.
Similarly, we know that .
From these two equations, we get and
.
Multiply the two equations to get . Solving, we get that
.
Solution 8
and
adding both we get
then we see
is 3/2 times
putting
as x we get x+3x=12 so x=3 and
= 3 so ab=
=
~ math_comb01
See also
1984 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 4 |
Followed by Problem 6 | |
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