log_4(64) = log_4(4^3) = 3.

log_4(x) = 1/2 implies x = 4^(1/2) = 2.

log_4(x) = 2 implies x = 4^2 = 16.

log_4(x) = 3 implies x = 4^3 = 64.

log_4(x) = 3 implies x = 4^3 = 64.

log_5(25) = 2 and log_5(5) = 1. Therefore, x = 2 + 1 = 3.

log_5(125) = log_5(5^3) = 3. Since log_5(25) = 2, we have x = 2, thus log_5(125) = 3.

log_5(5^x) = x.

log_5(25) = log_5(5^2) = 2.

log_5(x) = 1/2 implies x = 5^(1/2) = sqrt(5).

log_5(x) = 2 implies x = 5^2 = 25.

Since 49 = 7^2, log_7(49) = 2, thus x = 2.

From log_a(16) = 2, we have a^2 = 16, thus a = 4.

log_a(6) = log_a(2 * 3) = log_a(2) + log_a(3) = x + y.

log_a(6) = log_a(2 * 3) = log_a(2) + log_a(3) = x + y.

log_a(16) = log_a(4^2) = 2 * log_a(4) = 2 * 2 = 4.

log_a(4) = 2 implies a^2 = 4 => a = 2.

log_a(25) = log_a(5^2) = 2 log_a(5) = 2p, hence q = 2p.

log_a(25) = log_a(5^2) = 2 log_a(5) = 2p, hence q = 2p.

From the definition of logarithms, b = a^c.

log_a(bc) = log_a(b) + log_a(c) = p + q.

log_a(bc) = log_a(b) + log_a(c) = p + q.

log_b(27) = 3 implies b^3 = 27 => b = 3.

log_x(16) = 4 implies x^4 = 16 => x^4 = 2^4 => x = 2.

log_x(27) = 3 implies x^3 = 27 => x = 3.

log_x(4) = 2 implies x^2 = 4 => x = 2.

log_x(81) = 4 implies x^4 = 81 => x^4 = 3^4 => x = 3.

log_x(81) = 4 implies x^4 = 81 => x = 3.

M · N = 1*4 + 2*5 + 3*6 = 4 + 10 + 18 = 32.

M · N = 2*3 + 2*3 + 2*3 = 6 + 6 + 6 = 18.