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

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Answer 1 · 2016-08-09T10:58:57

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

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

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

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

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