State and explain the laws of logarithms with examples.

1. 1 · Open in two lines. A logarithm is the power to which a base must be raised to give a number: logₐ x = n means aⁿ = x. For example, log₂ 32 = 5 because 2⁵ = 32.
2. 2 · Product law. logₐ (mn) = logₐ m + logₐ n. Example: log 6 = log 2 + log 3.
3. 3 · Quotient law. logₐ (m/n) = logₐ m − logₐ n. Example: log 5 = log 10 − log 2 = 1 − 0.3010 = 0.6990.
4. 4 · Power law. logₐ (mⁿ) = n logₐ m. Example: log 8 = 3 log 2 = 0.9030.
5. 5 · Special values and change of base.
   • logₐ a = 1 and logₐ 1 = 0.
   • logₐ x = log x ÷ log a, e.g. log₄ 32 = log₂ 32 ÷ log₂ 4 = 5/2.
6. 6 · Use in competitive exams. Value questions, 'how many digits' problems and log equations in SSC and placement tests all rest on these laws.
7. 7 · Close in one line. The laws turn multiplication, division and powers into addition, subtraction and multiplication.

Scoring tip: Give a numerical example for every law; the examiner looks for the example as much as the formula.