I assume that by log( ) you mean loge( ) and not 
log10( ) as most people in the US mean by log( ).  
When we want to indicate loge we write ln( ). If
you meant log10 then tell me in the thank-you note 
form below and I'll get back to you. 

Use Euler's equation:

         ,

Substitute ln(1+i) for a+bi on the left 

         

         

eq. 1    

Set the real part on the left side of eq. 1 
equal to the real part on the right of eq. 1:

eq. 2    

Set the imaginary part on the left side of eq. 1 
equal to the imaginary part on the right of eq. 1:

eq. 3    

Divide equals by equals, that is, divide both side 
of eq. 2 by both sides of eq. 1





The tangent is positive in QI and QIII so



However in eq. 3

,

the left side is positive, and ea is positive,
so we can discard the QIII answer. 

Substitute in eq. 3, using 

















Square both sides to eliminate the square root:





Take ln of both sides:







So the solution is



Edwin