This calculates the log of a number and is independent of Math.log. This can be used with a big number or bigInt library to calculate a custom precision value of log.

log.js

//TODO: Still depends on Math.sqrt and Math.pow
var factorial = function(n) {
	++n;
	var r = 1;
  while(n-->1)
  	r*=n;
   return r;
};

var PI = function(n) {
	n = n || 30;
    var k = Math.sqrt(2)*2/9801;
    var sum = 0;
    for(var i=0; i<n; i++) {
          sum += (factorial(4*i)*(1103+26390*i))/
      (Math.pow(factorial(i), 4)*Math.pow(396, 4*i));
    }
    return 1/(sum*k);
};

var qlog = function(x, e=1e-16) {
    var a, b, c, d, de, ans, prev, k;
    a = x-1;
    b = x+1;
    c = (a*a)/(b*b);
    d = 2*a/b;
    ans = 0;
    k = 0;
    prev = 0;
    do {
        ans += (1/(2*k+1))*Math.pow(c, k);
        de = ans - prev;
        prev = ans;
        k++;
    }
    while(de >= e)
    return ans*d;
};

var log = function(x) {
    return ln(x)/ln(10);
};

//arithmatic mean
var M = function(x, y) {
    var a, b, d, e=1e-20;
    do {
        a = (x+y)/2;
        b = Math.sqrt(x*y);
        d = a-b;
        x = a;
        y = b;
    }
    while(d >= e)
    return x;
};

var calc_m = function(p, x) {
    var m = 1;
    var p2 = p/2;
    while(x*Math.pow(2, m) < Math.pow(2, p2)) 
        m++;

    return m;
};

var log = function(x, p=25) {
    var m, LN2, pi;
    LN2 = qlog(2);
    m = calc_m(p, x);
    pi = PI();
    return pi/(2*M(1, Math.pow(2, 2-m)/x))-m*LN2;
};