Given the equation $2 \log_y x = 3$, find the relationship between $x$ and $y$.

1. Problem Description

Given the equation $2 \log_y x = 3$, find the relationship between $x$ and $y$.

2. Solution Steps

We are given the equation $2 \log_y x = 3$.

First, divide both sides of the equation by 2:

$\log_y x = \frac{3}{2}$

We can rewrite this logarithmic equation in exponential form. The general formula for converting a logarithm to an exponential is:

$\log_b a = c \implies b^c = a$

Applying this to our equation $\log_y x = \frac{3}{2}$, we get:

$y^{\frac{3}{2}} = x$

To eliminate the fractional exponent, we can raise both sides of the equation to the power of 2:

$(y^{\frac{3}{2}})^2 = x^2$

$y^3 = x^2$

3. Final Answer

The relationship between $x$ and $y$ is $y^3 = x^2$.