Transformation and the error standard deviation
Transforming X does not affect the spread of response values at each value of X. Transformation of X therefore does not affect whether or not the linear model's assumption of constant error standard deviation holds.
However, transforming Y not only affects linearity of the relationship, but also affects whether or not the error standard deviation is constant.
Example
The raw data shown in the scatterplot on the top left shows both curvature and non-constant variance — the y-values are much more variable when X is near 0 than when X is high.
A log transformation of Y both linearises the relationship and removes the non-constant variance.
Fortunately, the same transformation of the response that linearises the relationship often also results in fairly constant error standard deviation.