How do you evaluate log_14 (-1)?

2 Answers
A. S. Adikesavan
Aug 9, 2016

((4n+1)pi)/ln 14 i, n = 0, +-1, +-2. +-3, ...

Explanation:

The values are complex.

log_14 (-1)

=ln(-1)/ln 14

=ln(i^2)/ln 14

=2 ln i /ln 14

=2lne^(i(2npi+pi/2))/ln 14, n = 0, +-1. +-2, +-3, ...

=((4n+1)pi i) /ln 14, n = 0, +-1. +-2, +-3, ..

Cesareo R.
Aug 9, 2016

log_{14}(-1) = i (pi pm 2kpi )/log_e 14

for k = 0,1,2, cdots

Explanation:

Let us investigate the complex solutions. We know by the logarithm definition

14^z = -1

Supposing now z = x + i y we have

14^x 14^{iy} = -1 so we have

14^x = 1-> x = 0 and
14^{iy} = -1

We know

14 = e^{log_e 14} so

14^{iy} = e^{i y log_e 14} = cos(y log_e14)+i sin(y log_e 14) = -1.

(We used de Moivre's identity e^{i phi)=cos(phi)+i sin(phi))

This condition is attained for

y log_e14 = pi pm 2kpi with k= 0,1,2,3,4,cdots

so

y = (pi pm 2kpi )/log_e 14

Finally

log_{14}(-1) = i (pi pm 2kpi )/log_e 14

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