Logarithm rules, explained with examples

A logarithm answers a question about powers: $\log_{a} b = c$ means exactly the same thing as $a^{c}=b$. Read it as "the power you raise $a$ to, to get $b$".

Every logarithm rule in Add Math is really an index law wearing a different notation, which is why this chapter sits right after indices and surds in the syllabus, the two topics share the same underlying rules.

A handful of facts follow directly from that definition and are worth knowing without working them out each time.

• $\log_{a} 1 = 0$, because $a^{0}=1$ for any base $a$.
• $\log_{a} a = 1$, because $a^{1}=a$.
• A logarithm is only defined for a positive number, and only for a positive base that is not equal to 1.

Four rules cover almost every logarithm manipulation in the syllabus. Three of them need to be memorised; the fourth, change of base, is provided on the exam formula list.

Product rule

The log of a product splits into a sum of logs, provided the base stays the same throughout.

$$\log_{a}(xy) = \log_{a} x + \log_{a} y$$

Quotient rule

The log of a quotient splits into a difference of logs, in the same order as the division.

$$\log_{a}\left(\dfrac{x}{y}\right) = \log_{a} x - \log_{a} y$$

Power rule

A power inside a logarithm moves out to the front as a multiplier.

$$\log_{a}(x^{n}) = n\log_{a} x$$

Change of base

This is the one rule given on the formula list, useful whenever a calculator or a simplification needs a different base than the one written in the question.

$$\log_{a} b = \dfrac{\log_{c} b}{\log_{c} a}$$

The four rules are simple individually, but they get mixed up under exam pressure in predictable ways.

• Treating $\log_{a}(x+y)$ as $\log_{a} x + \log_{a} y$. The product rule applies to multiplication, not addition, there is no rule that splits a log of a sum, and this is the single most common invented shortcut.
• Writing the quotient rule the wrong way round, subtracting the first log from the second instead of the other way around.
• Leaving the power $n$ inside the log instead of multiplying it out in front, especially when $n$ is a fraction or negative.
• Mixing bases when applying the product or quotient rule, the rule only works when every log in the expression shares the same base.

Example 1, simplify $\log_{2} 32 - \log_{2} 4$ without a calculator.

Apply the quotient rule first: $\log_{2} 32 - \log_{2} 4 = \log_{2}\left(\dfrac{32}{4}\right) = \log_{2} 8 = 3$, since $2^{3}=8$.

Example 2, solve $\log_{3} x + \log_{3}(x-2) = 1$ for $x>2$.

1. Combine the two logs using the product rule: $\log_{3}\left[x(x-2)\right] = 1$.
2. Convert to index form: $x(x-2) = 3^{1} = 3$.
3. Expand and rearrange: $x^{2}-2x-3=0$, which factorises to $(x-3)(x+1)=0$.
4. The two roots are $x=3$ and $x=-1$; reject $x=-1$ since the domain requires $x>2$, leaving $x=3$.

A non-programmable scientific calculator, the type used for both papers, usually has a $\log$ button for base 10 and an $\ln$ button for base $e$, but no direct way to evaluate a log in any other base. That is exactly where the change-of-base rule earns its place on the formula list, converting $\log_{a} b$ into base 10 or base $e$ before pressing any buttons.

For equations rather than pure simplification, the calculator is best used to check a final numeric answer, not to skip the algebra that gets you there.

Logarithm rules look abstract until you have applied each one a few times in context, and mixing up the product and quotient rule, or forgetting the domain check on an equation, is a very ordinary mistake to make while that familiarity is still building. A teacher who can watch which rule you reach for first, and why, can correct the mix-up before it becomes a habit.

Our teachers are experienced; lessons run online and are taught in English, while SPM papers themselves are set bilingually in Bahasa Melayu and English.

If logarithms, or indices and surds more broadly, feel shaky, a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience) is a reasonable place to start. We will not promise a particular grade, but we can help the four rules stop feeling interchangeable and start feeling like four distinct, reliable tools.