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# What is the natural logarithm of -1?

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What is the natural logarithm of -1?

Guest Sep 23, 2014

#1
+8

there are no natural logarithms for negative numbers.

the only way it will work is if you use an imaginary number.

Example:

$${{log}}_{{\sqrt{-{\mathtt{1}}}}}{\left(-{\mathtt{1}}\right)}$$ which is 2, since $${{\sqrt{-{\mathtt{1}}}}}^{\,{\mathtt{2}}} = -{\mathtt{1}}$$

Guest Sep 23, 2014
Sort:

#1
+8

there are no natural logarithms for negative numbers.

the only way it will work is if you use an imaginary number.

Example:

$${{log}}_{{\sqrt{-{\mathtt{1}}}}}{\left(-{\mathtt{1}}\right)}$$ which is 2, since $${{\sqrt{-{\mathtt{1}}}}}^{\,{\mathtt{2}}} = -{\mathtt{1}}$$

Guest Sep 23, 2014
#2
+26642
+5

With imaginary numbers there are other possibilities.  Since ei*pi = -1 then ln(-1) = ln(ei*pi) = i*pi

More generally we could have ln(-1) = n*i*pi where n is any odd integer.

Alan  Sep 24, 2014

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