# How to Solve a Logarithmic Practice Problem?

Solve the logarithmic practice problem

log x/(y – z) = log y/(z – x) = log z/(x – y) then find the value of x^{x} y^{y} z^{z}

## Solution to the logarithmic practice problem

Let, log x/(y – z) = log y/(z – x) = log z/(x – y) = k

log x/(y – z) = k

⇒ log x = k(y – z)

Multiply with x

x log x = kx(y – z)

⇒ log x^{x} = kxy – kxz……………eq(1)

log y/(z – x) = k

⇒ log y = k(z – x)

Multiply with y

y log y = ky(z – x)

⇒ log y^{y} = kyz – kxy……………eq(2)

log z/(x – y) = k

⇒ log z = k(x – y)

Multiply with z

z log z = kz(x – y)

⇒ log z^{z} = kxz – kyz……………eq(3)

Add equation 1, equation 2 and equation 3, then

log x^{x}+ log y^{y} + log z^{z} = kxy – kxz + kyz – kxy + kxz – kyz

⇒ log (x^{x} . y^{y} . z^{z}) = 0

⇒ x^{x} y^{y} z^{z} = 10^{0}

⇒ **x ^{x} y^{y} z^{z} = 1**