Using the logarithmic property (log_{b} b^n = n), we have (log_{10} 100 = log_{10} (10^2) = 2).

Rewrite the equation as (2^5 = x + 3), then solve for (x) to get (x = 31).

Using the logarithmic property (log_b (b^n) = n), we have (log_a (a^3 b^2) = log_a a^3 + log_a b^2 = 3 log_a a + 2 log_a b = 3 + 2 log_a b).

Using the logarithmic property (log_b (1/a) = - log_b a), we have (log_5 (1/125) = log_5 (5^{-3}) = -3).

Using the logarithmic property (log_b (a/c) = log_b a - log_b c), we have (log_a (b/c) = log_a b - log_a c).

Rewrite the equation as (3^2 = 2x - 1), then solve for (x) to get (x = 5).

Using the logarithmic property (log_b (b^n) = n), we have (log_a (a^2 b^3 c^4) = log_a a^2 + log_a b^3 + log_a c^4 = 2 log_a a + 3 log_a b + 4 log_a c = 2 + 3 log_a b + 4 log_a c).

Using the logarithmic property (log_b (1/a) = - log_b a), we have (log_2 (1/16) = log_2 (2^{-4}) = -4).

Using the logarithmic property (log_b (a/c) = log_b a - log_b c), we have (log_a (b^2 / c^3) = log_a b^2 - log_a c^3 = 2 log_a b - 3 log_a c).

Rewrite the equation as (4^3 = 3x + 2), then solve for (x) to get (x = 8).

Using the logarithmic property (log_b (b^n) = n), we have (log_a (a^5 b^2 c^3) = log_a a^5 + log_a b^2 + log_a c^3 = 5 log_a a + 2 log_a b + 3 log_a c = 5 + 2 log_a b + 3 log_a c).

Using the logarithmic property (log_b (1/a) = - log_b a), we have (log_3 (1/27) = log_3 (3^{-3}) = -3).

Using the logarithmic property (log_b (a/c) = log_b a - log_b c), we have (log_a (b^3 c^2 / d^4) = log_a b^3 + log_a c^2 - log_a d^4 = 3 log_a b + 2 log_a c - 4 log_a d).

Rewrite the equation as (5^2 = 4x - 3), then solve for (x) to get (x = 9).

Using the logarithmic property (log_b (b^n) = n), we have (log_a (a^7 b^4 c^5) = log_a a^7 + log_a b^4 + log_a c^5 = 7 log_a a + 4 log_a b + 5 log_a c = 7 + 4 log_a b + 5 log_a c).