MathsICSE Class 9 · Logarithms

Expanding, Combining and Evaluating Logarithms from Given Values (ICSE Class 9)

Updated 17 October 2026 · Fact-checked

To expand, turn products into sums, quotients into differences and powers into multiples. To combine, do the reverse into one log. To use given values such as log 2 and log 3, break the number into factors of 2, 3 and 10; for example log 12 = 2 log 2 + log 3 and log 5 = 1 − log 2.

Understand Expanding, Combining and Evaluating Logarithms from Given Values

Expanding. Apply the laws in order: powers, then products and quotients.
$ $lo g \frac{x y ^{2}}{z} = lo g x + 2 lo g y - lo g z, lo g \frac{a}{b} = \frac{1}{2} (lo g a - lo g b) .$ $

Combining. Move coefficients into powers, then add logs for products and subtract for quotients:
$ $2 lo g 3 + 3 lo g 2 - lo g 6 = lo g \frac{9 \times 8}{6} = lo g 12.$ $

Using given values. Suppose the question gives $lo g 2 = 0.3010$ and $lo g 3 = 0.4771$ (values supplied in the question; use exactly these). Break a number into $2$ , $3$ and $10$ :

$lo g 4 = 2 lo g 2 = 0.6020$ ; $lo g 6 = lo g 2 + lo g 3 = 0.7781$ .
$lo g 12 = 2 lo g 2 + lo g 3 = 1.0791$ ; $lo g 18 = lo g 2 + 2 lo g 3 = 1.2552$ .
$lo g 1.5 = lo g 3 - lo g 2 = 0.1761$ .
$lo g 5 = lo g \frac{10}{2} = 1 - lo g 2 = 0.6990$ .
$lo g 0.45 = lo g \frac{9 \times 5}{100} = 2 lo g 3 + (1 - lo g 2) - 2 = - 0.3468$ .

Digits of a power. If $lo g N = k$ , with $k$ having integer part $m$ , then $N$ has $m + 1$ digits. For $2^{20}$ : $20 lo g 2 = 6.0200$ , so $2^{20}$ has $7$ digits. For $3^{20}$ : $20 \times 0.4771 = 9.542$ , so $10$ digits. For $6^{10}$ : $10 (0.7781) = 7.781$ , so $8$ digits.

Accuracy. Use all given decimals and add exactly; do not round to fewer places in the middle. Give the answer to the same number of decimals as the data.

Level. Class IX is a school-exam class. The digit-counting application goes beyond routine and is at foundation-olympiad level; labelled hots.

Expanding, Combining and Evaluating Logarithms from Given Values: formulas to remember

Expansion: $lo g \frac{x y ^{2}}{z} = lo g x + 2 lo g y - lo g z$; Products, quotients, powers.
Combining: $n lo g m + lo g p = lo g (m^{n} p)$; Coefficients become powers.
Log of 5: $lo g 5 = 1 - lo g 2$; Since 5 = 10 ÷ 2.
Digits: $digits of N = ⌊ lo g N ⌋ + 1$; Integer part plus one.

How to solve Expanding, Combining and Evaluating Logarithms from Given Values questions

Factor into 2s, 3s and 10s, then apply the laws.

1Write the number (or fraction) as a product of 2, 3, 5 and 10 powers.
2Replace 5 by $10 \div 2$ so that only $lo g 2$ and $lo g 3$ are needed.
3Use the product, quotient and power laws.
4Substitute the given values and add or subtract carefully.
5For digits, find $lo g N$ and add 1 to its integer part.

Expanding, Combining and Evaluating Logarithms from Given Values in real life

Each concept, met in an everyday situation. Read the situation, try the problem, then check the solution. The one-line hook is what to remember.

Counting digits of a large number

Suppose a lab has $2^{20}$ possible codes and wants to know how many digits the number has (taking $lo g 2 = 0.3010$ ).

Find the number of digits.

Try it first, then show the solutionSolution

$lo g 2^{20} = 20 \times 0.3010 = 6.0200$ .
Integer part 6, so $6 + 1 = 7$ digits.

Answer: 7 digits (indeed $2^{20} = 1048576$ ).

Memory hook: Digits = integer part of the log plus 1.

Reading a number from its log

Imagine a gauge showing $lo g N = 2.6990$ for a count $N$ (hypothetical).

Find $N$ using $lo g 5 = 0.6990$ .

Try it first, then show the solutionSolution

$2.6990 = 2 + 0.6990 = lo g 100 + lo g 5 = lo g 500$ .
So $N = 500$ .

Answer: $N = 500$ .

Memory hook: Split the log into a whole number and a known part.

Tricks to remember Expanding, Combining and Evaluating Logarithms from Given Values

Short memory aids. Say each one aloud once, then use it on a question the same day.

Prime-ten factors
Build any number from 2, 3 and 10 (and 5 = 10/2).
Use it when: Using given log 2 and log 3.
Whole plus one
Digits = integer part of the log + 1.
Use it when: Counting digits.
Coefficient into power
n log m is log mⁿ when combining.
Use it when: Single-log answers.

Expanding, Combining and Evaluating Logarithms from Given Values: solved examples

Ordered from board level to HOTS. Try each one before you open the solution.

Example 1: Using a given value · 1 mark · MCQBoard level

Given $lo g 2 = 0.3010$ , the value of $lo g 4$ is:
(a) 0.1505
(b) 0.6020
(c) 0.9030
(d) 1.2040

Try it first, then show the solutionSolution

$lo g 4 = 2 lo g 2 = 0.6020$ .

Answer: (b) 0.6020

Example 2: Expanding · 3 marks · Short answerBoard level

Expand (i) $lo g \frac{x y ^{2}}{z}$ (ii) $lo g \frac{a}{b}$ .

Try it first, then show the solutionSolution

(i) $lo g x + 2 lo g y - lo g z$ .
(ii) $\frac{1}{2} lo g \frac{a}{b} = \frac{1}{2} (lo g a - lo g b)$ .

Answer: (i) log x + 2 log y − log z (ii) ½(log a − log b)

Example 3: Values from log 2 and log 3 · 4 marks · Short answerHarder board level

Given $lo g 2 = 0.3010$ and $lo g 3 = 0.4771$ , find (i) $lo g 24$ (ii) $lo g 36$ (iii) $lo g 0.2$ (iv) $lo g 15$ (v) $lo g 7.5$ .

Try it first, then show the solutionSolution

(i) $lo g 24 = 3 lo g 2 + lo g 3 = 0.9030 + 0.4771 = 1.3801$ .
(ii) $lo g 36 = 2 lo g 2 + 2 lo g 3 = 0.6020 + 0.9542 = 1.5562$ .
(iii) $lo g 0.2 = lo g \frac{2}{10} = 0.3010 - 1 = - 0.6990$ .
(iv) $lo g 15 = lo g 3 + lo g 5 = 0.4771 + (1 - 0.3010) = 1.1761$ .
(v) $lo g 7.5 = lo g \frac{15}{2} = 1.1761 - 0.3010 = 0.8751$ .

Answer: 1.3801; 1.5562; −0.6990; 1.1761; 0.8751

Example 4: Single log and value · 4 marks · Short answerHarder board level

Express $2 lo g 3 + 3 lo g 2 - lo g 6$ as a single logarithm and find its value using $lo g 2 = 0.3010$ , $lo g 3 = 0.4771$ .

Try it first, then show the solutionSolution

$= lo g 9 + lo g 8 - lo g 6 = lo g \frac{9 \times 8}{6} = lo g 12$ .
Value $= 2 (0.3010) + 0.4771 = 1.0791$ .
Check directly: $0.9542 + 0.9030 - 0.7781 = 1.0791$ . ✓

Answer: $lo g 12 = 1.0791$

Example 5: How many digits? · 5 marks · Long answerHOTS / competency

Using $lo g 2 = 0.3010$ and $lo g 3 = 0.4771$ , find the number of digits in $2^{20}$ , $3^{20}$ and $6^{10}$ .

Try it first, then show the solutionSolution

$lo g 2^{20} = 6.0200$ : integer part 6, so 7 digits.
$lo g 3^{20} = 20 \times 0.4771 = 9.5420$ : 10 digits.
$lo g 6^{10} = 10 (0.7781) = 7.7810$ : 8 digits.
Check: $2^{20} = 1048576$ (7 digits), $3^{20} = 3486784401$ (10 digits), $6^{10} = 60466176$ (8 digits).

Answer: 7, 10 and 8 digits.

Where students lose marks in Expanding, Combining and Evaluating Logarithms from Given Values

The slips examiners see most often, and what to write instead.

Mistake: Using log 5 as a separate unknown
It must be written through log 2.
Wrong working
$lo g 5 = 0.5$ .
Right working
$lo g 5 = 1 - 0.3010 = 0.6990$ .
Do this: Use 5 = 10/2.
Usually costs: Whole answer.
Mistake: Digits equal to the integer part
Add one.
Wrong working
$lo g N = 6.02 \Rightarrow 6$ digits.
Right working
$7$ digits.
Do this: Integer part plus one.
Usually costs: Final answer.
Mistake: Dropping the coefficient when combining
n log m is log mⁿ.
Wrong working
$2 lo g 3 = lo g 3$ .
Right working
$2 lo g 3 = lo g 9$ .
Do this: Move coefficients to powers.
Usually costs: Whole answer.

Exam tips

Class IX is examined in school papers; the values of log 2 and log 3 are given, never recalled.
Use exactly the given decimals; do not round in the middle.
Write the factorisation of the number before substituting.
State the law used (product, quotient, power) beside each step.

Practice questions on Expanding, Combining and Evaluating Logarithms from Given Values

Write each answer on paper first. Open the hint only if you are stuck, and the answer only when you have finished.

Q1 · 1 mark · MCQBoard level
Given $lo g 2 = 0.3010$ , the value of $lo g 5$ is:
(a) 0.6020
(b) 0.6990
(c) 0.7781
(d) 1.3010
Hint
$lo g 5 = 1 - lo g 2$ .
Answer
(b) 0.6990
Q2 · 3 marks · Short answerHarder board level
Express $\frac{1}{2} lo g 16 + lo g 5 - lo g 2$ as a single logarithm and evaluate it.
Hint
$\frac{1}{2} lo g 16 = lo g 4$ .
Answer
$lo g 4 + lo g 5 - lo g 2 = lo g \frac{20}{2} = lo g 10 = 1$ .
Q3 · 4 marks · Short answerHarder board level
Given $lo g 2 = 0.3010$ and $lo g 3 = 0.4771$ , find $lo g 0.075$ .
Hint
$0.075 = \frac{3 \times 25}{1000}$ .
Answer
$lo g 3 + 2 lo g 5 - 3 = 0.4771 + 2 (0.6990) - 3 = 0.4771 + 1.3980 - 3 = - 1.1249$ .
Q4 · 1 mark · Assertion-reasonHarder board level
Assertion (A): $lo g 8 = 3 lo g 2$ .
Reason (R): $lo g m^{n} = n lo g m$ .
(a) Both (A) and (R) are true and (R) explains (A).
(b) Both are true but (R) does not explain (A).
(c) (A) is true, (R) is false.
(d) (A) is false, (R) is true.
Hint
$8 = 2^{3}$ .
Answer
(a)
Q5 · 4 marks · Case-basedHOTS / competency
Case study: A software system produces $2^{20}$ possible codes. Take $lo g 2 = 0.3010$ and $lo g 3 = 0.4771$ .
(a) Find the number of digits in $2^{20}$ .
(b) Find the number of digits in $3^{20}$ .
(c) Find the number of digits in $6^{10}$ and say which is larger of $3^{20}$ and $6^{10}$ .
Hint
Digits = integer part of the log + 1.
Answer
(a) 7 (b) 10 (c) 8; since log 3²⁰ = 9.542 > 7.781 = log 6¹⁰, 3²⁰ is larger.

Test your recall

Answer in your head or on paper first, then open the card. Pulling an answer out of memory makes it stick far better than re-reading.

1.Write log 12 using log 2 and log 3.Show answer
2 log 2 + log 3.
2.Write log 5 using log 2.Show answer
1 − log 2.
3.How do you find the number of digits of N?Show answer
Integer part of log N plus 1.
4.Combine 3 log 2 + log 5.Show answer
log 40.

Expanding, Combining and Evaluating Logarithms from Given Values: frequently asked questions

Do I need to memorise log 2 and log 3?

No; questions give the values.

Why 1 − log 2 for log 5?

Because 5 = 10 ÷ 2 and log 10 = 1.

Is the digit count in the syllabus?

It is an application of the laws of logarithms and is included here as an advanced item.