Laws of Logarithms

Product rule: log(x·y) = log x + log y, a product in the argument becomes a sum. Quotient rule: log(x/y) = log x − log y, a fraction becomes a difference. Power rule: log(x^r) = r·log x, the exponent moves in front of the logarithm as a factor. All three hold in every base (ln, lg, log₂) and for positive arguments. Number example with the base-10 logarithm: lg(1000·100) = lg 1000 + lg 100 = 3 + 2 = 5, and indeed 10⁵ = 100000 = 1000·100 ✓. The laws are the translation of the laws of exponents into the logarithm world: because exponents add when powers are multiplied, logarithms add, since a logarithm IS an exponent.

Take logarithms on both sides, because the power rule brings the x down from the exponent: 2ˣ = 10 becomes x·ln 2 = ln 10, so x = ln 10/ln 2 ≈ 2.303/0.693 ≈ 3.32. Check: 2^3.32 ≈ 9.98 ≈ 10 ✓. Which base you choose does not matter, ln and lg give the same x; you can also write the result directly as log₂ 10. The same pattern solves all growth and decay tasks: 1.03ᵗ = 2 (doubling at 3% interest) gives t = ln 2/ln 1.03 ≈ 23.4 years. Important: first isolate the power, then take logarithms. For 5·2ˣ = 40, first divide by 5 (2ˣ = 8, x = 3) rather than pulling apart log(5·2ˣ) without respecting the rules.

The change-of-base formula reads log_b x = ln x/ln b (or with lg instead of ln, the quotient is the same). You need it whenever your calculator does not offer the desired base: you enter log₂ 10 as ln 10/ln 2 ≈ 3.32. It can be derived in two lines: from b^y = x, taking logarithms gives y·ln b = ln x, so y = ln x/ln b. The formula also shows that all logarithm systems differ only by a constant factor; that is why logarithm curves of different bases look like vertically stretched copies of each other. Most common error: swapping numerator and denominator. Mnemonic: the base goes at the bottom of the fraction, just as it sits at the bottom in the symbol log_b.

Because the product law only translates products into sums, not sums into sums. The logarithm answers the question about the exponent, and exponents add when powers are MULTIPLIED (b^m·b^n = b^(m+n)), not when they are added. A number test destroys the misconception immediately: lg(10 + 10) = lg 20 ≈ 1.301, but lg 10 + lg 10 = 2. There simply is no general simplification law for log(x + y); such expressions stay as they are or require other techniques like factoring (lg(50 + 50) = lg 100 = 2 only because you add first). The same warning applies to log(x − y). In exams, wrongly splitting sums is one of the most frequent point losses in logarithm tasks.

They are logarithms to different bases. ln is the natural logarithm with base e ≈ 2.718; it is the standard in calculus, because only it has the clean derivative 1/x. lg is the base-10 logarithm; it fits orders of magnitude and sits inside pH (pH = −lg[H₃O⁺]), decibels and earthquake magnitudes. log₂ is the base-2 logarithm of computer science: memory sizes, search trees, the number of halving steps in binary search. All three obey the same laws of logarithms and differ only by constant factors, convertible via change of base: log₂ x = ln x/ln 2 ≈ 1.443·ln x. Beware of notation: a bare "log" means lg (engineering), ln (university mathematics) or log₂ (computer science) depending on the field; when in doubt state the base.