Find the General Value of log(-3)(-2)

Solution:

Let log_(-3)(-2) = x + iy

Using logarithmic properties:

(x + iy) = log_e(-2) / log_e(-3)
(x + iy) log_e(-3) = log_e(-2)
(x + iy) log' 3 + i(2m + 1)πy = log 2 + i(2n + 1)π

By equating real and imaginary parts, we get:

x log 3 - y(2m + 1)π = log 2
y log 3 + x(2m + 1)π = (2n + 1)π

Solving these equations, we obtain:

x = [(2m + 1)(2n + 1)π² + (log 2)(log 3)] / [(log 3)² + (2m + 1)²π²]
y = [log 3(2n + 1)π - (2m + 1)π log 2] / [(log 3)² + (2m + 1)²π²]

Conclusion

Thus, the general value of log_(-3)(-2) is given by the above expressions for x and y.

Source: Manonmaniam Sundaranar University, Directorate of Distance & Continuing Education.