The number of integer solutions of the equati | vedclass

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(A) Given the equation $|1-i|^x=2^x$. First,calculate the modulus of the complex number $1-i$: $|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{1+1} = \sqrt{2} =

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(A) Given the equation $|1-i|^x=2^x$. First,calculate the modulus of the complex number $1-i$: $|1-i| = \sqrt{1^2 + (-1)^2} = \sqrt{1+1} = \sqrt{2} = 2^{\frac{1}{2}}$. Substitute this into the equation: $(2^{\frac{1}{2}})^x = 2^x$. $2^{\frac{x}{2}} = 2^x$. Since the bases are equal,equate the exponents: $\frac{x}{2} = x$. $x = 2x \Rightarrow x = 0$. Thus,there is only $1$ integer solution,which is $x=0$.

The number of integer solutions of the equation $|1-i|^x=2^x$ is

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