The properties of logarithms are rules that simplify the process of working with logarithmic expressions. The main properties include the product property, which states that the logarithm of a product is the sum of the logarithms of the factors: log_b(m * n) = log_b(m) + log_b(n). The quotient property indicates that the logarithm of a quotient is the difference of the logarithms: log_b(m / n) = log_b(m) - log_b(n). Another important property is the power property, which states that the logarithm of a number raised to an exponent is the exponent times the logarithm of the base: log_b(m^k) = k * log_b(m). These properties help in solving equations and simplifying expressions involving logarithms, making them essential tools in mathematics.