Question

if (2.3)x=(0.23)y=1000 then find the value of 1/x-1/y

Answer 1

Answer is 1/3,

Here is the solution :

(2.3)^x = 10³,
=> 2.3 = ((10)^(3/x))
=> 0.23 = (((10)^(3/x))/(10))) --------(1)

(0.23)^y = 10³
=> 0.23 = ((10)^(3/y)) -------(2),

=> (1) and (2) are equal since 0.23 = 0.23,

=> 10^((3/x)-(1)) = 10^(3/y)
=> Bases are equal so powers are equal,

=> (3/x) - 1 = 3/y
=> (3/x) - (3/y) = 1,
=> 1/x - 1/y = 1/3,

Therefore : Hence Proved !

Hope you understand, Have a Great day !
Thanking you, Bunti 360 !!

Answer 2

(2.3)^x=1000 and (0.23)^y=1000

log(2.3)^x =log1000 and log(0.23)^y=1000

xlog(2.3)=log10^3 and log(0.23)^y =log10^3

xlog(2.3) = 3log10 and ylog(0.23) =3log10

xlog(2.3) = 3×1 and ylog(0.23)=3  [log10=1]

⇒log(2.3)=3/x and log0.23 =3/y

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3/x -3/y =log2.3  -log(0.23)

3(1/x - 1/y) =log(2.3/0.23)

⇒1/x - 1/y ={log(10)}/3

⇒1/x-1/y =(1)/3

⇒1/x -1/y =1/3