Question

Given that 3^x = 4^y = 12^z, show that z = (xy)/(x+y).

Answer 1

`3^(x) = 4^(y) = 12^(z)`

`3^(x) = 4^(y) = (4 \cdot 3)^(z)`

`3^(x) = 4^(y) = 4^(z) \cdot 3^(z)`


`\text{Let a } = 3^(x) = 4^(y) = 4^(z) \cdot 3^(z)`

`log_3a = x`

`log_4a = y`

`log_((4 \cdot 3))a = z`

Using change of base:

`x = (lna)/(ln3)`

`y = (lna)/(ln4)`

`z = (lna)/(ln(4 \cdot 3))`


`ln3 = (lna)/(x)`

`ln4 = (lna)/(y)`

`ln(4 \cdot 3) = (lna)/(z)`

Now, ln(4 · 3) = ln(4) + ln(3)


`(lna)/(z) = (lna)/(x) + (lna)/(y)`

`(1)/(z) = (1)/(x) + (1)/(y)`

`(1)/(z) = (x + y)/(xy)`


`\therefore z = (xy)/(x + y)`