LogarithmYou are currentlynot logged in Click here to log in 

The logarithm to base $b$ of a number $x,$ written $y=log_b(x),$ is the number $y$ such that $b^y=x.$ It is the inverse function to exponentiation.
When restricted to real number, logarithms are only defined on positive numbers. Logarithms are never defined at zero, since $x^y$ is never 0 (except in the trivial case where $x=0$ and $y$ is nonzero).
There are some standard rules for logarithms, similar to, and derived from, the rules for exponents and indices. Thus we have:
When ranging over complex numbers the logarithm is a multivalued function (and hence not a function at all).
Consider the complex number $z.$ Because of the polar representation of a complex number we can write $z~=~r.e^{i\theta}.$ However, we can add $2\pi$ to $\theta$ and get the same result, so we also have $z~=~r.e^{i(\theta+2\pi)}.$ Thus $r.e^{i\theta}~=~r.e^{i(\theta+2\pi)}.$
Taking logarithms base $e$ of both sides we have:
The error is in assuming that even if $x=y$ it must then follow that $log(x)=log(y).$ This is not a valid deduction if $log$ is not a single valued function  it's akin to assuming that just because $x^2=y^2$ then $x=y,$ or that if $\sin(x)=\sin(y)$ then $x=y.$
However, if we consider logarithms on an appropriate Riemann Surface then the analogous function is a true, singlevalued function.
Last change to this page Full Page history Links to this page 
Edit this page (with sufficient authority) Change password 
Recent changes All pages Search 