The wrong value on the Y-axis when using lethargy units

Here is the lecture for learning,
13_Neutronics_2023_NEA.pdf (3.2 MB)
on page 11,
the value on the Y-axis is: x*df/dx,
but the X-axis is represented as log10(x), so the value on the Y-axis should be: ln(10)xdf/dx

This is only an arbitrary scaling factor that does not change the shape of the curve, which the lethargy representation is intended for.
As for absolute values to handle, what matters is anyway dN/dE, with its well-defined units.