What is the mathematical relation for the change in water density at different temperatures?
To do this one must measure density accurately at many temperatures and then fit a curve to the measurements. This has been done by several investigators in several industries. For example in the sugar industry we have

$$\rho =(((((-281.03006e-12*t +105.84601e-9)*t-46.241757e-6)*t-7.9905127e-3)*t+16.952577)*t +999.83952)/(1+16.887236e-3*t)$$

where $t$ is the temperature in °C and the density $g·cc^-1$. The coefficients were taken from the ICUMSA formula with the sugar concentrations set to 0.

And 

$$\rho = 0.99984+t*(6.7715e-05-t*(+9.0735e-06-t*(1.015e-07-t*(+1.3356e-09-t*(1.4421e-11-t*(+1.0896e-13-t*(4.9038e-16-9.7531e-19*t)))))))$$

returns the density of water, also in $g·cc^-1$,  as a function of centigrade temperature (ITS 1990). It is based on a fit to data from
Bettin, H.; Spieweck,F.: "Die Dichte des Wassers als Funktion der Temperatur nach Einführung der Internationalen
Temperaturskala von 1990. PTB-Mitt. 100 (1990) pg 195-196
The referenced data set lists densities for each 0.1°C over the range (0,100). The data used in determining this
polynomial are the subset on integer degree values. Sporadic checking shows that the fit appears to be
accurate to 1 count in the 6th decimal place. The rms residual wrt the fit points is 3.1E-7 and the peak
residual 5.5E-7. The residuals appear noiselike.

You ought to be able to paste those formulas into a visualization program and plot values. I don't see any logarithmic behaviour. I've left the * in the formulas so you can do that. Note, when reading them, that 16.887236e−3∗t means (16.887236e−3)∗t.