Question

The value of $ {{\log }_{1-i}}\left( 1+i \right) $ is
(a) $ \dfrac{\log \sqrt{2}+i\dfrac{\pi }{4}}{\log \sqrt{2}-i\dfrac{\pi }{4}} $
(b) $ \dfrac{\log \sqrt{2}-i\dfrac{\pi }{4}}{\log \sqrt{2}+i\dfrac{\pi }{4}} $
(c) $ \dfrac{\log 2+i\dfrac{\pi }{4}}{\log 2-i\dfrac{\pi }{4}} $
(d) None of these

Answer 1

Hint: We will use the change of base rule and rewrite the given expression. We will look at the definition of a modulus and argument of a complex number. Then we will use the formula for logarithm of a complex number, which is $ \log z=\log \left( |z| \right)+\arg \left( z \right)i $ . To use this, we will find the modulus and the argument of the two complex numbers involved and obtain the correct answer.

Complete step by step answer:
The change of base rule states that we have $ {{\log }_{b}}a=\dfrac{\log a}{\log b} $ with a common base for both the logarithms on the right hand side. Hence, we can rewrite the given expression in the following manner,
 $ {{\log }_{1-i}}\left( 1+i \right)=\dfrac{\log \left( 1+i \right)}{\log \left( 1-i \right)} $
with a common base for the logarithms on the right hand side.
Let $ z=x+iy $ be a complex number. Then, we define the modulus of the complex number to be $ |z|=\sqrt{{{x}^{2}}+{{y}^{2}}} $ and the argument of the complex number, denoted by $ \arg \left( z \right) $, is the angle of the vector representing the complex number from the positive real axis in a complex plane. If this angle is measured counter-clockwise, then it is positive. Otherwise, it is negative.
The complex numbers involved in the logarithm given are $ {{z}_{1}}=1+i $ and $ {{z}_{2}}=1-i $ . We will calculate the modulus of both these numbers in the following manner,
 $ \begin{align}
  & |{{z}_{1}}|=\sqrt{{{1}^{2}}+{{1}^{2}}} \\
 & \therefore |{{z}_{1}}|=\sqrt{2} \\
\end{align} $
Similarly,
 $ \begin{align}
  & |{{z}_{2}}|=\sqrt{{{1}^{2}}+{{\left( -1 \right)}^{2}}} \\
 & \therefore |{{z}_{2}}|=\sqrt{2} \\
\end{align} $
In the complex plane, the number $ {{z}_{1}}=1+i $ is represented by the vector joining the origin to the point $ \left( 1,1 \right) $ . This vector makes an angle of $ \dfrac{\pi }{4} $ with the positive real axis in the counter-clockwise direction. Hence, $ \arg \left( {{z}_{1}} \right)=\dfrac{\pi }{4} $ . Similarly, the number $ {{z}_{2}}=1-i $ is represented by the vector joining the origin to the point $ \left( 1,-1 \right) $ . This vector makes an angle of $ \dfrac{\pi }{4} $ with the positive real axis in the clockwise direction. Hence, $ \arg \left( {{z}_{1}} \right)=-\dfrac{\pi }{4} $ .
We know the formula for logarithm of a complex number is $ \log z=\log \left( |z| \right)+\arg \left( z \right)i $ .
Using the above formula and substituting the respective values of the modulus and argument, we have
 $ \log \left( 1+i \right)=\log \sqrt{2}+\dfrac{\pi }{4}i $
 $ \log \left( 1-i \right)=\log \sqrt{2}-\dfrac{\pi }{4}i $
Now, substituting these values in the expression formed after changing the base of the logarithm, we get
 $ {{\log }_{1-i}}\left( 1+i \right)=\dfrac{\log \sqrt{2}+\dfrac{\pi }{4}i}{\log \sqrt{2}-\dfrac{\pi }{4}i} $
Therefore, the correct option is (a).

Note:
 It is important to understand the components of a complex number, that is the modulus and the argument. The logarithm function is a multi-valued function. Therefore, the concept of principal value applies to this function as well. The concept of principal value for logarithm involves the argument component of a complex number.