Laws of Logarithms | SPM Additional Mathematics

What this method is for

A logarithm answers the question 'to what power must I raise the base to get this number?' The laws of logarithms let you rewrite a complicated logarithm as a combination of simpler ones.

A product inside the log becomes a sum, a quotient becomes a difference, and a power comes out to the front as a multiplier. In Add Math you use these laws to express one logarithm in terms of others, to write several logarithms as a single logarithm, and to simplify an expression before solving an equation.

These three laws, together with the change-of-base rule and two small identities, cover almost every logarithm manipulation on the paper.

product law

\log_a(xy) = \log_a x + \log_a y

quotient law

\log_a\frac{x}{y} = \log_a x - \log_a y

power law

\log_a x^{n} = n\log_a x

change of base

\log_a b = \frac{\log_c b}{\log_c a}

Two identities finish the toolkit: $\log_a a = 1$ and $\log_a 1 = 0$ .

When to reach for it

Reach for the laws whenever a logarithm hides a product, a quotient, or a power inside its argument, or whenever a question gives you the logarithms of a few small numbers and asks you to build the logarithm of a larger one. Phrases such as 'express in terms of $p$ and $q$ ', 'write as a single logarithm', or 'evaluate without a calculator' are direct invitations to use them.

If two logarithms in the same expression carry different bases, convert them to a common base first with the change-of-base rule, otherwise the product and quotient laws do not apply. Deciding which law fits each part before you write anything keeps the working short and every step easy to credit.

The steps

1
Check the bases
Confirm every logarithm shares the same base. If not, use $\log_a b = \frac{\log_c b}{\log_c a}$ to bring them to one base.
2
Factorise each argument
Write each number as a product, quotient or power of the numbers whose logarithms you already know, for example $45 = 3^{2}\times 5$ .
3
Split products into sums
Apply $\log_a(xy)=\log_a x+\log_a y$ to break a product into separate logarithms.
4
Split quotients into differences
Apply $\log_a\frac{x}{y}=\log_a x-\log_a y$ to any division.
5
Bring powers to the front
Use $\log_a x^{n}=n\log_a x$ so each exponent becomes a coefficient.
6
Use the identities
Replace $\log_a a$ with $1$ and $\log_a 1$ with $0$ , then substitute the given values and simplify.

Worked example

Q1[4 marks]

Given that $\log_2 3 = p$ and $\log_2 5 = q$ , express $\log_2\frac{45}{8}$ in terms of $p$ and $q$ .

Show worked solution

First apply the quotient law to separate the top from the bottom.

\log_2\frac{45}{8} = \log_2 45 - \log_2 8

Factorise $45$ into numbers whose logarithms are known: $45 = 9\times 5 = 3^{2}\times 5$ . Apply the product law.

\log_2 45 = \log_2\left(3^{2}\times 5\right) = \log_2 3^{2} + \log_2 5

Bring the power to the front with the power law:

\log_2 3^{2} = 2\log_2 3 = 2p, \qquad \log_2 5 = q

So $\log_2 45 = 2p + q$ . Now handle $\log_2 8$ .

Since $8 = 2^{3}$ , the identity $\log_2 2 = 1$ gives

\log_2 8 = \log_2 2^{3} = 3\log_2 2 = 3

Combine the two parts:

\log_2\frac{45}{8} = (2p + q) - 3 = 2p + q - 3

So $\log_2\frac{45}{8} = 2p + q - 3$ . Every number was rewritten using only $3$ , $5$ and the base $2$ , which is exactly what 'in terms of $p$ and $q$ ' asks for.

Common pitfalls

Writing $\log_a(x+y)=\log_a x+\log_a y$ . The laws split products and quotients, never sums or differences inside the argument.
Applying the power law to only part of the argument, the exponent must belong to the whole thing being logged.
Combining logarithms with different bases without first using the change-of-base rule.
Confusing $\frac{\log_a x}{\log_a y}$ with $\log_a\frac{x}{y}$ ; the first is a ratio of logs, the second is the log of a quotient.
Forgetting the identities, so a term like $\log_2 8$ is left as a logarithm instead of simplified to $3$ .

How a teacher helps

The single most common slip we see is a student splitting $\log_a(x+y)$ as though addition inside the log behaved like multiplication. In one-to-one lessons our teachers watch the exact line where you choose a law and stop you the moment the wrong one is applied, so the habit is corrected before it hardens.

We also drill the small factorisation step, spotting that $45 = 3^{2}\times 5$ or that $8 = 2^{3}$ , because that is where most marks are won or lost. Every teacher on spmaddmath.com.my is experienced.

Lessons are online and taught in English, paced to how quickly the laws are clicking for you.

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Frequently asked questions

Do the laws of logarithms work for any base?

Yes. The product, quotient and power laws hold for any valid base $a>0$ , $a\neq 1$ , as long as every logarithm in the step uses the same base.

If bases differ, convert first with $\log_a b = \frac{\log_c b}{\log_c a}$ .

Why is

\log_a(x+y)

not equal to

\log_a x + \log_a y

Because the laws only rearrange products, quotients and powers, not sums. $\log_a x + \log_a y$ actually equals $\log_a(xy)$ , not $\log_a(x+y)$ .

There is no law that simplifies the logarithm of a sum.

When do I need the change-of-base rule?

Use it when an expression or equation mixes logarithms of different bases, or when you need to evaluate a logarithm your calculator does not have a direct key for. Rewriting everything in one base lets the other laws apply.

Are the laws of logarithms on the SPM formula list?

You should memorise the product, quotient, power and change-of-base rules, treat them as core knowledge rather than relying on the formula sheet. Because SPM Add Math uses analytic marking, showing each law you apply earns method marks even if the final value slips.

Source:SRC-DSKP-EN

Written by the spmaddmath.com.my editorial team.· Last updated 5 September 2026

How to use the laws of logarithms