Solve simultaneous logarithmic equations with different bases? How do I solve these simultaneous equations? 2log_x y+2 log_y x=5 xy=8

Buszmenan 2022-10-11 Answered
Solve simultaneous logarithmic equations with different bases?
How do I solve these simultaneous equations?
2 l o g x y + 2 l o g y x = 5
x y = 8
I've tried to convert the first formula to fraction form and continue from there, but I can't seem to get anywhere. I've tried to do
x = 8 / y
and substitute to the first equation, but I still can't seem to solve this. How do I go about in solving these types of equations?
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Answers (1)

Samantha Braun
Answered 2022-10-12 Author has 9 answers
Say log x y = a
Therefore
a + 1 a = 5 2
2 a 2 5 a + 2 = 0
( 2 a 1 ) ( a 2 ) = 0
a = 2 , 1 2
Hence we have
log x y = 2 , 1 2
Now
x y = 8
1 + log x y = log x 8
Hence
log x 8 1 = 2
log x 8 = 3
log 8 x = 1 3
x = 8 1 3 = 2
Similarly we also have
log x 8 1 = 1 2
log x 8 = 3 2
x = 4
The solutions hence follow.
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