Cambridge O Level Additional Mathematics · Syllabus 4037 · Logarithmic and Exponential Functions
Laws of Logarithms
What is Laws of Logarithms?
The three rules that convert operations inside a logarithm into simpler operations outside it, valid for a base a with a greater than zero and a not equal to one and for strictly positive arguments M and N. The product law says the logarithm of M times N equals the logarithm of M plus the logarithm of N. The quotient law says the logarithm of M divided by N equals the logarithm of M minus the logarithm of N. The power law says the logarithm of M raised to the power k equals k times the logarithm of M. All three are index laws in disguise, because a logarithm is an exponent, and none of them applies to a sum: the logarithm of M plus N cannot be split.
This definition is part of the Logarithmic and Exponential Functions chapter in Cambridge O Level Additional Mathematics.
Questions students ask about Laws of Logarithms
Why can't you split \(\log(M+N)\) into \(\log M+\log N\)?
Because the laws of logarithms are index laws in disguise, and there is no index law for \(a^{p}+a^{q}\): a sum of powers does not simplify. Products become sums, \(\log_a(MN)=\log_a M+\log_a N\); quotients become differences; powers become multipliers, \(\log_a(M^{k})=k\log_a M\). A sum inside the logarithm has no law at all. One check settles it: \(\lg(1+99)=\lg100=2\), but \(\lg1+\lg99\approx1.996\).