The natural logarithm of a complex number is calculated by the following equation :
The Excel Imln function returns the natural logarithm of a supplied complex number.
The format of the function is:
where the inumber argument is a complex number.
Note that complex numbers are simply stored as text in Excel. When a text string in the format "a+bi" or "a+bj" is supplied to one of Excel's builtin complex number functions, this is interpreted as a complex number.
Also the complex number functions can accept a simple numeric value, as this is equivalent to a complex number whose imaginary coefficient is equal to 0.
Therefore, the inumber argument can be supplied to the Excel Imln function as either:
  A simple number  
or  
  A complex number encased in quotation marks  eg. "5+3i"  
or  
  A reference to a cell containing a complex number or a numeric value  
or  
  A value returned from another Excel function or formula 
These are all included in the examples below.
The spreadsheet below shows 4 different examples of the Imln function.
The format of the functions are shown in the spreadsheet on the left and the resulting values are shown in the spreadsheet on the right.
Formulas:

Results:

Note that, in cell B1 of the above spreadsheet, the supplied inumber argument is the integer 2, which is equal to the complex number 2+0i.
Further information and examples of the Excel Imln function can be found on the Microsoft Office website
If you get an error from the Excel Imln Function, this is likely to be one of the following:
#NUM!    Occurs if the supplied inumber argument is not recognised as a complex number 
#NAME?   
Occurs when Analysis ToolPak addin is not enabled in your Excel. + Show How to Do This in Excel 2003: 