Question

X^y=y^x and x=2y prove that y=2.

Answer 1

`{x}^(y) = {y}^(x) \: - - - (1)`

`x = 2y \: - - - (2)`

Substitute (2) into (1):

`{2y}^(y) = {y}^(2y)`

Log both sides:

`lg(2 {y}^(y) ) = lg(y^(2y) )`

`y \: lg(2y) = 2y(lg \: y)`

`y \: lg(2) + y \: lg(y) = 2y \: lg(y)`

`y \: lg(2) = 2y \: lg(y) - y \: lg(y)`

`y \: lg(2) = y \: lg(y)`

Divide by y on both sides:

`lg(2) = lg(y)`

`{10}^(lg(2)) = {10}^(lg(y))`

2= y

y= 2 (proved)

Step-by-step explanation:

• step 5: lg(m)^n= nlg(m)

• Step 6: lg(mn)= lg(m) +lg(n)