/// <summary>
/// Compute the natural logarithm of x to a given scale, x > 0.
/// </summary>
public static DecimalX Ln(this DecimalX x, int scale)
{
// Check that scale > 0.
if (scale <= 0)
{
throw new ArgumentException(InvalidScale2);
}
// Check that x > 0.
if (x.Signum() <= 0)
{
throw new ArgumentException(NegativeOrZeroNaturalLog);
}
// The number of digits to the left of the decimal point.
var magnitude = x.ToString().Length - -x.Exponent - 1;
if (magnitude < 3)
{
return(lnNewton(x, scale));
}
// x^(1/magnitude)
var root = x.IntRoot(magnitude, scale);
// ln(x^(1/magnitude))
var lnRoot = lnNewton(root, scale);
// magnitude*ln(x^(1/magnitude))
var a = DecimalX.Create(magnitude);
var result = a.Multiply(lnRoot);
return(DecimalX.Rescale(result, -scale, RoundingMode.HalfEven));
}