Question based on exponential functions

Question

I need a little bit of help in the following exercise:

If $a^x=b^y=c^z$ and $b^2=ac$, then prove that $y=\frac{2xz}{x+z}$.

Answer 1

Let $$a^x=b^y=c^z=k$$

$$a=k^{\frac 1x}$$

$$b=k^{\frac 1y}$$

$$c=k^{\frac 1z}$$

Now given, $$b^2=ac$$

$$\left(k^{\frac 1y}\right)^2 = k^{\frac 1x} \cdot k^{\frac 1z}$$

$$k^{\frac 2y} = k^{\frac 1x + \frac 1z}$$

$$\frac 2y = \frac 1x + \frac 1z$$

$$\frac 2y = \frac {z+x}{xz}$$

Reciprocating both sides,

$$\frac y2 = \frac {xz}{x+z}$$

$$y = \frac {2xz}{x+z}$$

Answer 2

from the given system we get $$a=e^{\frac{\ln(t)}{x}}$$ and so on thus we have $$e^{\frac{2\ln(t)}{y}}=e^{\frac{\ln(t)}{x}}\cdot e^{\frac{\ln(t)}{z}}$$ since $$t\ne 1$$ we get $$\frac{2}{y}=\frac{1}{z}+\frac{1}{x}$$ p.s.: $t=a^x$ and so on