helo sir/madam

Question

can u help me to solve this problem

solve the following simultaneous equation

logx Y+logy X=5/2

xy=64

Answer 1

I saw this one pass by me before.
Are you distinguishing between "x" and "X"?
If so, you have 4 variables with only 2 equations.

Answer 2

I will assume they are the same x

recall that log_a b = log b/log a

so log_xy = logy/logx
and log_yx = logx/logy

so your equation can be written as:

logy/logx + logx/logy = 5/2

let logx/logy = a
then we have

1/a + a = 5/2
times 2a
2 + 2a^2 = 5a
2a^2 - 5a + 2 = 0
(a - 2)(2a - 1) = 0
a = 2 or a = 1/2

if a = 2
logx/logy = 2
log_yx = 2
y = x^2

then in xy=64
x(x^2) = 64
x^3 = 64
x = 4 ------------> so x = 4 , y = 16

if a = 1/2
log_yx = 1/2
x = y^(1/2) = √y
in xy=64
.....
x = 16, y = 4