Using stellar spectroscopy to measure stellar parameters, why is it $\log g$? Stellar spectroscopy can in principle measure the stellar surface gravity, radii, effective temperature, and stellar rotation. 
Why is it that surface gravity $g$ is always measured as $\log g$? 
How exactly is this measured?
 A: Answer to: Why people report the surface gravities as log g instead of g?
Typical values range from above $g = 10^4 = 10,000$ cm/s$^2$ or so for main sequence stars to $10^1 = 10$  cm/s$^2$ for giants.  That's quite a large range.   More compact and convenient to write $\log g$.
Answer to: How is $\log g$ measured?
If you imagine two hypothetical stars that have the same surface parameters (i.e. same temperature, composition), except that their $\log g$ is different, then their spectral lines would look different.
The star with high $\log g$ is is more massive or more compact. The pressure is higher. This affects the atoms at the surface of the star an leads to a broadening of the lines.   The reverse is also true.
In practice astronomers use fancy computer simulations that theoretically predict how stellar spectra look like for wide range of choices of $\log g$ and temperature and compositions etc.  They then use a computer to quickly compare the observed spectra against these models to determine which one is the closest match.
A: In addition to fitting the line profiles of lines that are sensitive to pressure-broadening (and hence gravity), particularly hydrogen Balmer and alkali metal lines, as per Selma de Mink's answer; there is also the ionisation balance technique.
This technique is essentially to demand that the chemical abundance determined from lines due to transitions in species of the same element, but with different ionisation states, are the same.
For example, since a star has only one iron abundance, we demand that the iron abundance we determine from the strength of absorption lines due to neutral iron (Fe I) and singly ionised iron (Fe II) are the same. This is accomplished by adjusting both the temperature and surface gravity, but since temperature can also be constrained from other techniques, this gives the gravity.
The basic physics behind this is the Saha equation for the equilibrium ionisation fraction in a plasma. The fraction of ionised atoms increases with temperature but decreases with electron density. At fixed temperature this means that there is less ionisation in a high pressure, high gravity atmosphere.
Why $\log g$? Basically because of the very wide range of gravities encountered in stellar astrophysics. Indeed, if we include white dwarfs at the upper end and supergiants at the lower end, the range is something like $10^0$ to $10^8$ cm/s$^2$, reflecting that stars of quite similar masses can have a factor of 10,000 range in radii.
It is also the case that changes in spectra are generally quite subtle. Big changes in gravity are generally needed to produce measurable changes, so the fitting parameter tends to be $\log g$ rather than $g$.
