Question

If a^x=b^y and a^y=b^x, show that x=y.​

Answer 1

`▪▪▪▪▪▪▪▪▪▪▪▪▪  {\huge\mathfrak{Answer}}▪▪▪▪▪▪▪▪▪▪▪▪▪▪`

`\large \boxed{ \mathfrak{Step\:\: By\:\:Step\:\:Explanation}}`

Let's solve ~

Using first equation we get,

• 
  `{a}^(x) = {b}^(y)`

`a = \sqrt[x]{ {b}^(y) }`

• 
  `a = {b}^{ (y)/(x) }`

now, plug the value of a in other equation ~

• 
  `a {}^(y) = b {}^(x)`

• 
  `( {b}^{ (y)/(x) } ) {}^(y) = {b}^(x)`

• 
  `{(b) }^{ (y)/(x) * y} = {b}^(x)`

• 
  `{(b)}^{ \frac{ {y}^(2) }{x} } = {b}^(x)`

• 
  `\frac{ {y}^(2) }{x} = x`

• 
  `{y}^(2) = {x}^{} * x`

• 
  `{y}^(2) = {x}^(2)`

• 
  `y = x`

I hope it helps ~