Difference between revisions of "2023 AMC 12A Problems/Problem 19"
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<cmath>\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023</cmath> | <cmath>\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023</cmath> |
Revision as of 01:19, 10 November 2023
Problem
What is the product of all solutions to the equation
Solution 1
For , transform it into
. Replace
with
. Because we want to find the product of all solutions of
, it is equivalent to finding the sum of all solutions of
. Change the equation to standard quadratic equation form, the term with 1 power of
is canceled. By using Vieta, we see that since there does not exist a
term,
and
.
~plasta
Solution 2 (Same idea with Solution 1)
Rearranging it give us:
let be
, we get
by veita's formula,
~lptoggled
See also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 18 |
Followed by Problem 20 |
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 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.