Transformations and the error standard deviation
In a scatterplot of Y against X, transforming X moves the crosses horizontally, but does not affect the spread of response values at each value of X.
If the error standard deviation is the same for each x in a plot of Y against X, it will also be constant in a plot of Y against any transformation of X.
Transformation of X therefore does not affect whether or not the linear model's assumption of constant error standard deviation holds.
However,
Transformation of the response, Y, not only affects linearity of the relationship, but also affects whether or not the error standard deviation is constant.
This is more easily explained in an example than with words.
Prices of second-hand Mazda cars
The scatterplot below shows the retail prices of 124 Mazda cars, obtained from the newspaper The Melbourne Age on 8 February 1992. The grey line is the least squares line fitted to the data.
The residual plot on the right highlights two problems. Firstly there is clearly nonlinearity — the prices level off at high ages. Also, the standard deviation of the price is much lower for cars over 10 years old — there is non-constant error standard deviation.
Drag the red line on the vertical axis upwards to apply a power transformation to the price. Observe that a log transformation (between a power of 0.01 and -0.01) both linearises the relationship reasonably well and also gives residuals with fairly constant spread. (Use the arrow keys on the keyboard to make fine adjustments to the power.)
Fortunately, the same transformation of the response that linearises the relationship often also results in fairly constant error standard deviation.