Making sense of logarithms in Add Math

Most students meet the word "logarithm" and quietly panic, because it sounds far more advanced than it is. The truth is friendlier: a logarithm is just an exponent, a power, asked about from the other direction.

Every logarithm is really asking one plain question: what power do I raise the base to, to get this number?

So $\log_{2} 8 = 3$ simply says "the power you raise $2$ to, in order to get $8$, is $3$", because $2^{3} = 8$. The little number written below, here $2$, is called the base.

Read every log that way and it stops being a mystery symbol and becomes a question you can answer.

$$\log_{a} x = y \quad\Longleftrightarrow\quad a^{y} = x$$

That double-headed arrow is the single most important line in the whole chapter. If you can flip fluently between the two sides, the log form and the index form, most log problems become index problems you already know how to handle.

In the Add Math syllabus, logarithms sit in the same chapter as indices and surds, and that is not an accident. Indices and logarithms are two views of a single relationship between three numbers, a base, a power, and a result.

Indices start from the base and the power and ask for the result. Logarithms start from the base and the result and ask for the power.

Because they describe the same relationship, they are inverses of each other. Two facts follow immediately from the definition and are worth knowing on sight:

• $\log_{a} 1 = 0$, because anything to the power $0$ is $1$.
• $\log_{a} a = 1$, because a base to the power $1$ is itself.

Keeping the index picture in mind is what stops the log laws feeling arbitrary. Every log law below is really an index law you already trust, translated into log language.

There are three laws that do almost all the work, and each mirrors an index law. Because multiplying powers of the same base adds their exponents, multiplying inside a log adds the logs:

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

In the same way, dividing inside a log subtracts, because dividing powers subtracts their exponents:

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

And a power inside a log comes out to the front as a multiplier, this is the law that quietly does the heavy lifting when you solve equations:

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

There is one more tool you will lean on: change of base. Your calculator is a non-programmable scientific calculator, and it only has a $\log$ button for base $10$ (and a natural log).

To evaluate a log in another base, rewrite it:

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

For instance, $\log_{2} 5 = \dfrac{\log_{10} 5}{\log_{10} 2}$, which your calculator can now handle directly.

Here is where logs earn their place in the paper. Suppose you need to solve $3^{x} = 20$.

The unknown is stuck up in the exponent, so ordinary algebra cannot reach it. Taking the logarithm of both sides frees it, because the power law brings the $x$ down to the front:

$$\log 3^{x} = \log 20 \;\Rightarrow\; x \log 3 = \log 20 \;\Rightarrow\; x = \frac{\log 20}{\log 3}$$

From there the calculator finishes the job. Notice the shape of it: the log did not "solve" the equation by magic, it turned a problem you could not touch into division you can.

That is the pattern behind almost every log question in Add Math.

Most lost marks in this chapter come from a small number of tempting-but-wrong moves. Knowing them in advance is worth more than any amount of extra practice on the parts you already do correctly.

• $\dfrac{\log x}{\log y}$ is not $\log x - \log y$. Subtraction comes from $\log\!\left(\dfrac{x}{y}\right)$; a fraction of two separate logs is just change of base.
• The power law needs the power to sit on the whole argument: $\log x^{2}$ becomes $2\log x$, but $(\log x)^{2}$ does not, those are different objects.
• You can only take the log of a positive number, so always check that any answer keeps the inside of every log positive.

Logarithms are a chapter that clicks the moment the index picture behind them clicks, and stays confusing for as long as they are memorised as loose rules. Working one-to-one, a teacher can watch how you read a log, catch the exact place the index connection is missing, and rebuild it so the laws feel obvious rather than arbitrary.

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

If logs are a sticking point, you are welcome to begin with a one-hour paid trial class at the teacher's own rate (from RM50 per hour, depending on experience). We won't promise a grade, but we can help you turn logarithms from a source of dread into one of the more dependable marks on your paper.