Evaluate log base 2 of -16

Problem

(log_2)(−16)

Solution

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

2. Observe the argument of the given expression. The argument is −16 which is a negative number.

3. Determine the existence of a real solution. Since the argument is less than zero, there is no real number y such that 2=−16

4. Consider complex numbers if necessary. In the complex plane, the logarithm of a negative number involves the imaginary unit i and the natural logarithm.

5. Apply the change of base formula using the natural logarithm: (log_2)(−16)=ln(−16)/ln(2)

6. Evaluate the natural log of a negative number using the identity ln(−x)=ln(x)+i*π for x>0

7. Substitute the values: ln(−16)=ln(16)+i*π

8. Simplify the expression: (ln(16)+i*π)/ln(2)=(ln(2)+i*π)/ln(2)=(4*ln(2)+i*π)/ln(2)=4+(i*π)/ln(2)

Final Answer

(log_2)(−16)=4+(i*π)/ln(2)

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