Evaluate natural log of -6

Problem

ln(−6)

Solution

1. Identify the domain of the natural logarithm function for real numbers. The natural logarithm ln(x) is only defined for x>0 in the set of real numbers.

2. Apply the definition of the complex logarithm. For a negative real number, we use the formula ln(z)=ln(z)+i*arg(z)

3. Determine the magnitude and the argument. For z=−6 the magnitude |z| is 6

4. Find the principal argument. The number −6 lies on the negative real axis of the complex plane, so its principal argument Arg*(−6) is π

5. Substitute these values into the complex logarithm formula to find the principal value.

Final Answer

ln(−6)=ln(6)+i*π

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