There is a sense in which logarithms demonstrate that addition is structually equivalent to multiplication. Imagine a mirror that reflects multiplication into a logarithm world. When you multiply, a x b, and a and b are reflected as log(a) and log(b). The multiplication is reflected as addition. The result of the multiplication is reflected. Log(a x b) = log(a) + log(b).

People have trouble remembering that log(1)=0. Think of it this way. 1 is the mutiplicative identity. 1 x a = a for all a. The log reflection maps 1 to the addiitive identity in log world. 0 is the additive identity. a + 0 = 1 for all a. So log(1) = 0. The identity in one world translates to the identity in the other one.

1/a is the multiplicative inverse of a, since a x (1/a) = 1, the multiplicative identity. Similarly, -a is the additive inverse, so log(1/a) = -log(a).