Polar equation and Cartesian equation

Question

For the polar equation,

$r \sin \theta = \ln r + \ln (\cos\theta)$

Is that equivalent to $y = \ln x $ ?

Answer 1

We know IF we want to change from cartesian coordinates $(x,y)$ to polar coordinates $(r, \theta)$ , we must make the substitution $x = r \cos \theta $ and $y = r \sin \theta $. Notice,

$$ \ln r + \ln cos \theta = \ln (r \cos \theta ) = \ln x $$

Hence, $r \sin \theta = y = \ln x $