A linear pot has a resistive element of constant cross-section, resulting in a device where the resistance between the wiper and one end terminal is proportional to the distance between them. Linear describes the electrical 'law' of the device, not the geometry of the resistive element.
A log pot has a resistive element that either 'tapers' in from one end to the other, or is made from a material whose resistivity varies from one end to the other. This results in a device where output voltage is a logarithmic (or inverse logarithmic depending on type) function of the mechanical angle of the pot.
Most (cheaper) "log" pots are actually not logarithmic, but use two regions of different, but constant, resistivity to approximate a logarithmic law. A log pot can also be simulated with a linear pot and an external resistor. True log pots are significantly more expensive.