What has been done here in this step?

$x = e^{\log_5 y}$

Using change of base:

$\log_5 y = \frac{\ln y}{\ln 5}$

Therefore:

$x = e^{\frac{\ln y}{\ln 5}}$

Rewrite the exponent:

$x = \left(e^{\ln y}\right)^{\frac{1}{\ln 5}}$

Since:

$e^{\ln y} = y$

So:

$x = y^{\frac{1}{\ln 5}}$

Now:

$\frac{1}{\ln 5} = \log_5 e$

Thus finally:

$x = y^{\log_5 e}$

Change of base identity is used inside the exponents!

It is known as exponential-logarithm identity

The identity is