Solve this equation with p&p to find two nice solutions.
72xยท48y=6xy
The person who shared this problem on Twitter and posted a video of their solution on YouTube just found these two solutions. A bunch of people who replied on Twitter just found these two solutions.
However, if you use your favorite graphing calculator, you will see there are many more solutions.
Find a set of parametric equations that will generate all solutions.
Wolfram Alpha gives the solution of
solve 78^x*48^y=6^(x*y) for x, y
as
x = (y log(48) + 2 i p n)/(y log(6) - log(78))
and y log(6) != log(78)
and n element Z
ans shows a graph of the real solutions which does show a horizontal asymptote at y=log(78)/log(6), matching that prohibition on y.
clc
for y=-4:.1:7
x=(y*log(48) )/(y*log(6)-log(78));
fprintf('%16.10f %4.1f\n',x,y)
end
shows a limited portion of the graph:
x y
1.3437299901 -4.0
1.3308290165 -3.9
1.3175139878 -3.8
1.3037646454 -3.7
1.2895593871 -3.6
1.2748751539 -3.5
1.2596873053 -3.4
1.2439694808 -3.3
1.2276934479 -3.2
1.2108289324 -3.1
1.1933434306 -3.0
1.1752020001 -2.9
1.1563670270 -2.8
1.1367979652 -2.7
1.1164510454 -2.6
1.0952789481 -2.5
1.0732304362 -2.4
1.0502499410 -2.3
1.0262770949 -2.2
1.0012462012 -2.1
0.9750856337 -2.0
0.9477171514 -1.9
0.9190551175 -1.8
0.8890056041 -1.7
0.8574653641 -1.6
0.8243206456 -1.5
0.7894458215 -1.4
0.7527017981 -1.3
0.7139341610 -1.2
0.6729710060 -1.1
0.6296203907 -1.0
0.5836673271 -0.9
0.5348702157 -0.8
0.4829565969 -0.7
0.4276180609 -0.6
0.3685041162 -0.5
0.3052147582 -0.4
0.2372914066 -0.3
0.1642057755 -0.2
0.0853461056 -0.1
-0.0000000000 0.0
-0.0926671584 0.1
-0.1936395913 0.2
-0.3040862184 0.3
-0.4254061156 0.4
-0.5592880937 0.5
-0.7077897949 0.6
-0.8734441978 0.7
-1.0594052896 0.8
-1.2696508079 0.9
-1.5092699587 1.0
-1.7848807826 1.1
-2.1052508623 1.2
-2.4822471660 1.3
-2.9323383775 1.4
-3.4790648855 1.5
-4.1572910319 1.6
-5.0209450692 1.7
-6.1581127007 1.8
-7.7231686073 1.9
-10.0135841656 2.0
-13.6857429726 2.1
-20.5300438707 2.2
-37.7819102433 2.3
-164.4808605478 2.4
78.8818538190 2.5
33.3430380376 2.6
21.7283488217 2.7
16.4178619288 2.8
13.3745152166 2.9
11.4018749916 3.0
10.0194263757 3.1
8.9967681246 3.2
8.2096166714 3.3
7.5850201326 3.4
7.0773372937 3.5
6.6565511454 3.6
6.3021102245 3.7
5.9994701840 3.8
5.7380484334 3.9
5.5099612063 4.0
5.3092147897 4.1
5.1311711506 4.2
4.9721851587 4.3
4.8293523851 4.4
4.7003300571 4.5
4.5832075533 4.6
4.4764111523 4.7
4.3786329105 4.8
4.2887768280 4.9
4.2059175900 5.0
4.1292685846 5.1
4.0581568529 5.2
3.9920032780 5.3
3.9303067786 5.4
3.8726315934 5.5
3.8185969730 5.6
3.7678687648 5.7
3.7201524962 5.8
3.6751876545 5.9
3.6327429283 6.0
3.5926122281 6.1
3.5546113393 6.2
3.5185750952 6.3
3.4843549768 6.4
3.4518170673 6.5
3.4208403000 6.6
3.3913149538 6.7
3.3631413538 6.8
3.3362287479 6.9
3.3104943299 7.0
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Posted by Charlie
on 2023-06-23 12:55:00 |
computer findings
