Laws of Logarithms

A logarithm is an exponent in disguise: $\log_a y = x$ means $a^x = y$. The laws follow from index laws, and 0606 tests whether you can drive them in both directions without inventing laws that don’t exist.

The laws (same base throughout)

• $\log a + \log b = \log ab$
• $\log a - \log b = \log\left(\dfrac{a}{b}\right)$
• $n \log a = \log a^n$
• $\log_a a = 1$ $\cdot$ $\log_a 1 = 0$ $\cdot$ change of base: $\log_b a = \dfrac{\log a}{\log b}$ (any new base, useful form: $\log_b a = \dfrac{1}{\log_a b}$)

All memorised, never given.

The direction decision

Condense (many logs → one) when solving an equation: get a single log on each side, then drop the logs or apply the definition. Expand (one log → pieces) when using given values:

Given $\log 2 = p$ and $\log 3 = q$, express $\log 12$ in terms of $p$ and $q$: $\log 12 = \log(4 \times 3) = \log 2^2 + \log 3 =$ $2p + q$

The skill is factorising the number into the given primes, $\log 12$ questions are factorisation questions wearing logs.

Condensing to solve: $\log_5 x + \log_5 (x - 4) = 1$ $\log_5[x(x - 4)] = 1 \to x(x - 4) = 5 \to x^2 - 4x - 5 = 0 \to x = 5$ or $x = -1$ $x = -1$ rejected (log of a negative undefined) $\to$ $x = 5$

The condensing line (M), the definition applied (M), the rejection with reason (B/A): the standard log-equation anatomy.

The fake laws, the syllabus’s most-marked misconceptions

• $\log(a + b)$ $\ne$ $\log a + \log b$
• $\log a \times \log b$ $\ne$ $\log ab$, and $\dfrac{\log a}{\log b} \ne \log\left(\dfrac{a}{b}\right)$; the quotient of logs is change of base, not a subtraction
• $(\log a)^n$ $\ne$ $n \log a$, the power must be on the argument, not the log

Examiner reports cite these every session. When tempted, translate to indices and check with small numbers: $\log 10 + \log 10 = 2$, but $\log 20 \ne 2$.

Common mistakes

• Fake laws above
• Laws applied across different bases (all laws require a common base)
• $n \log a$ direction confused when condensing ($3 \log 2 = \log 8$, not $\log 6$)
• Numbers not factorised into the given values’ primes
• Solutions of condensed equations unchecked against positive-argument requirements

Full topic context: Logs & Exponentials notes.