Add logarithms using the rule: log A + Log B = Log (AB). This follows from the definition: If Log A = a, 10^a = A. If Log B = b, 10^b = B. If Log (AB) = k, 10^k = AB. So 10^k = AB = 10^a X 10^b = 10^(a + b) or k = a + b, so Log (AB) = Log A + Log B. To find the logarithm of a product you only need to add the logarithms of the factors. Instead of multiplying two numbers, look up the logarithms, add the logarithms, look up the number whose logarithm is that sum. Multiplication is reduced to addition -- and table look up.
Subtract logarithms using the rule: log A - Log B = Log (A/B). This follows from the definition: If Log A = a, 10^a = A. If Log B = b, 10^b = B. If Log (A/B) = n, 10^n = A/B. So 10^n = A/B = 10^a / 10^b = 10^(a - b) or n = a - b, so Log (A/B) = Log A + Log B. Instead of dividing two numbers, look up the logarithms, subtract the logarithms, look up the number whose logarithm is that difference. Division is reduced to addition and table usage.
Combine numbers and logarithms by using the exponent rule: Log (A^k) = k Log A. This is easy to prove: Log (A^k) = Log (A * A * A* ...k times) = Log A + Log A + ...k times = k Log A. This reduces exponentiation to multiplication. To find a number raised to a power, look up the logarithm of the number, multiply it by the power, then find the answer that has this logarithm.