I am curious to the relation of the density of water at different temperature and as to also how one would come to an equation that took the temperature of the water and would output it's density? I have graphed density vs temp from tabulated values and it seems to look logarithmic, but wanted to make sure. I am interested in this as I am currently taking analytic chemistry.
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.
$$\rho = 0.99984+t*(6.7715e-05-t*(+9.0735e-06-t*(1.015e-07-t*(+1.3356e-09-t*(1.4421e-11-t*(+1.0896e-13-t*(4.9038e-16-9.7531e-19*t)))))))$$
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.