Question

How do you evaluate `log_4 (-1/64)`?

Answer 1

`log_4(-1/64) = -3+i (pi pm 2kpi )/log_e 4`

for `k = 0,1,2,cdots`

Explanation:

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

`4^z = -1/64 = -4^{-3}`

Supposing now `z = x + i y` we have

`4^x 4^{iy} = -4^{-3}` so we have

`4^x = 4^{-3}->x = -3` and
`4^{iy} = -1`

We know

`4 = e^{log_e 4}` so

`4^{iy} = e^{i y log_e 4} = cos(y log_e 4)+i sin(y log_e 4) = -1`. This condition is attained for

`y log_e4 = pi pm 2kpi` with `k= 0,1,2,3,4,cdots`

so

`y = (pi pm 2kpi )/log_e 4`

Finally

`log_4(-1/64) = -3+i (pi pm 2kpi )/log_e 4`