Take a look at this log implementation.

This is from fdlibm that has the implementations (following the IEEE-754) of a lot of math functions in C for humans.

From the implementation:

Method

1. Argument Reduction: find k and f such that

    x = 2^k * (1+f),
    where sqrt(2)/2 < 1+f < sqrt(2) .

2. Approximation of log(1+f).

Let s = f/(2+f) ; based on log(1+f) = log(1+s) - log(1-s)
     = 2s + 2/3 s**3 + 2/5 s**5 + .....,
         = 2s + s*R

• We use a special Reme algorithm on [0,0.1716] to generate polynomial of degree 14 to approximate R The maximum error of this polynomial approximation is bounded by 2**-58.45. In other words,

                 2      4      6      8      10      12      14
    R(z) ~ Lg1*s +Lg2*s +Lg3*s +Lg4*s +Lg5*s  +Lg6*s  +Lg7*s
    (the values of Lg1 to Lg7 are listed in the program)

and

    |      2          14          |     -58.45
    | Lg1*s +...+Lg7*s    -  R(z) | <= 2 
    |                             |

Note that 2s = f - s*f = f - hfsq + s*hfsq, where hfsq = f*f/2. In order to guarantee error in log below 1ulp, we compute log by

    log(1+f) = f - s*(f - R)    (if f is not too large)
    log(1+f) = f - (hfsq - s*(hfsq+R)). (better accuracy)

3. Finally,

     log(x) = k*ln2 + log(1+f).  
            = k*ln2_hi+(f-(hfsq-(s*(hfsq+R)+k*ln2_lo)))

• Here ln2 is split into two floating point number:

        ln2_hi + ln2_lo,

where n*ln2_hi is always exact for |n| < 2000.

Real implementation and special cases of the explanation you can check in this link.