KBAT: Multi-Step Logarithm Problems

Q: How do I spot a hidden quadratic in a logarithm question?

Look for a repeated expression. If a block like log⁡2x\log_2 xlog2x appears twice, let uuu stand for it. Equations such as u+1u=52u+\frac{1}{u}=\frac{5}{2}u+u1=25 become quadratics once you clear the fraction, and factorising finishes the job.

Q: How is Add Math Paper 2 marked on these questions?

Paper 2 is 2 hours 30 minutes and 100 marks, and marking is analytic, method marks are awarded line by line. Changing base, substituting, solving the quadratic and checking validity each earn credit, so set out every step.

What makes this a KBAT question

A routine logarithm question applies one law once, expand $\log ab$ , or solve $\log_3(2x+1)=2$ . A multi-step KBAT question chains several ideas and hides the structure: it may combine the product, quotient or power laws with a change of base, and often conceals a quadratic behind a substitution.

That is higher-order thinking, Kemahiran Berfikir Aras Tinggi: each law is familiar, but you must decide which to use, in what order, and recognise a hidden form, for example that $\log_x 2$ equals $\tfrac{1}{\log_2 x}$ . Nothing on the page announces 'this is a quadratic'.

In Add Math this rewards students who see the shape of an equation, not only its surface.

One worked problem, in the style of Paper 2

This is an original question written in the style of SPM Paper 2. Attempt it before reading the solution.

Q1[6 marks]

Solve the equation $\log_2 x + \log_x 2 = \dfrac{5}{2}$ , giving each value of $x$ in exact form.

Show worked solution

Understand. The equation mixes two different bases: $\log_2 x$ has base $2$ , while $\log_x 2$ has base $x$ .

They cannot be added directly. We need one variable, so the first job is to write both terms with the same base.

Plan. Use the change-of-base law $\log_x 2=\dfrac{\log_2 2}{\log_2 x}=\dfrac{1}{\log_2 x}$ .

Let $u=\log_2 x$ ; the equation becomes $u+\dfrac{1}{u}=\dfrac{5}{2}$ , a hidden quadratic. Solve for $u$ , then recover $x$ , and check both answers are valid.

Execute and check. Substituting $u=\log_2 x$ :

u+\frac{1}{u}=\frac{5}{2}

Multiply every term by $2u$ (note $u\neq 0$ , since $\log_2 x=0$ would make $\log_x 2$ undefined):

2u^{2}+2=5u \;\Rightarrow\; 2u^{2}-5u+2=0

Factorise the quadratic:

(2u-1)(u-2)=0 \;\Rightarrow\; u=\tfrac{1}{2} \;\text{or}\; u=2

Return to $x$ through $u=\log_2 x$ , so $x=2^{\,u}$ :

u=\tfrac{1}{2}\Rightarrow x=2^{1/2}=\sqrt{2}\, \qquad u=2\Rightarrow x=2^{2}=4

Both values are valid, since a logarithm base must be positive and not equal to $1$ : $\sqrt{2}>0$ , $\sqrt{2}\neq 1$ , and $4>0$ , $4\neq 1$ . So $x=\sqrt{2}$ or $x=4$ .

Check. For $x=4$ : $\log_2 4=2$ and $\log_4 2=\tfrac{1}{2}$ , giving $2+\tfrac{1}{2}=\tfrac{5}{2}$ .

For $x=\sqrt{2}$ : $\log_2\sqrt{2}=\tfrac{1}{2}$ and $\log_{\sqrt{2}}2=2$ , giving $\tfrac{1}{2}+2=\tfrac{5}{2}$ . Both satisfy the original equation.

Finding a sensible first step

When a logarithm question looks tangled, the reliable first step is to make everything the same base. Scan for terms in different bases or a variable sitting in the base, and use the change-of-base law to bring them together, that alone untangles most multi-step problems.

Next, look for a repeated block. If the same expression, such as $\log_2 x$ , appears more than once, let a single letter stand for it; very often the equation then reveals itself as a quadratic you can factorise.

Only after that should you solve. Finally, plan to check: logarithms carry hidden conditions, the argument must be positive and a base must be positive and not $1$ , so you may need to reject an answer that breaks them.

What markers reward

Marking is analytic, so method marks are awarded line by line. On a multi-step logarithm question a marker looks for:

Correct use of the change-of-base law to put all terms in one base, for example $\log_x 2=\frac{1}{\log_2 x}$ .
A clear substitution such as $u=\log_2 x$ , which exposes the hidden quadratic.
The quadratic formed correctly, $2u^{2}-5u+2=0$ , and solved by factorisation or formula.
Both values of the substitution carried back to $x$ , not just one.
Answers left in exact form, including surds like $\sqrt{2}$ , rather than rounded decimals.
A validity check that each $x$ keeps every logarithm defined, with any invalid root rejected.

How a teacher helps

The block that stops students is rarely the algebra, it is seeing that a term hides a reciprocal or that a substitution reveals a quadratic. In a one-to-one lesson our teachers train that eye: we practise 'same base first', then hunt for the repeated block to substitute, so the moves become automatic.

We also drill the final validity check, because a lost answer or an un-rejected root costs marks the working had already earned. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

Message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.

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Frequently asked questions

How do I handle two different logarithm bases in one equation?

Use the change-of-base law to rewrite everything in one base. A handy special case is $\log_x 2=\frac{\log_2 2}{\log_2 x}=\frac{1}{\log_2 x}$ , which turns a mixed-base equation into a single-variable one.

How do I spot a hidden quadratic in a logarithm question?

Look for a repeated expression. If a block like $\log_2 x$ appears twice, let $u$ stand for it.

Equations such as $u+\frac{1}{u}=\frac{5}{2}$ become quadratics once you clear the fraction, and factorising finishes the job.

Do I need to check my logarithm answers?

Yes. A logarithm's argument must be positive, and a base must be positive and not $1$ .

After solving, test each value against these conditions and reject any that break them, but keep valid surd answers like $\sqrt{2}$ in exact form.

How is Add Math Paper 2 marked on these questions?

Paper 2 is 2 hours 30 minutes and 100 marks, and marking is analytic, method marks are awarded line by line. Changing base, substituting, solving the quadratic and checking validity each earn credit, so set out every step.

Source:SRC-DSKP-EN SRC-FORMAT

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