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CBSE · Class 9 · Mathematics

Logarithms (Advanced)

Standard difficulty · 20 questions · 40 marks · Time: 32 minutes

Name: Date: Score: ____ / 40

Show your working for Sections B and C. Answers are on the last page — try every question first!

Section A — Choose the correct answer8 × 1 = 8 marks
1.
The value of log2 16 is:
  1. −4
  2. 5
  3. 8
  4. 4
2.
log5 1375 + log5 5 − log5 11 equals:
  1. 8
  2. 3
  3. 4
  4. 5
3.
logₐ 1 equals (a > 0, a ≠ 1):
  1. a
  2. undefined
  3. 1
  4. 0
4.
log3 5 × log5 243 equals:
  1. 15
  2. 6
  3. 5
  4. 4
5.
If log5 2 = p and log5 3 = q, then log5 20 equals:
  1. 3p + 1
  2. 2q + 1
  3. 2p + 2
  4. 2p + 1
6.
On the Richter scale, an earthquake of magnitude 7 compared with one of magnitude 5 is:
  1. 7/5 times as strong
  2. 2 times as strong
  3. 20 times as strong
  4. 100 times as strong
7.
pH = −log10 [H+]. How many times as acidic is a solution of pH 4 as one of pH 6?
  1. 1,000
  2. 2
  3. 100
  4. 20
8.
The solution of log3 (5x + 66) = 4 is:
  1. x = 3
  2. x = 4
  3. x = 2
  4. x = 5
Section B — Short answer8 × 2 = 16 marks
9.
Write as a single logarithm: 3 + log2 14.
10.
Find the exact value of log25 5.
11.
Solve log7 x + log7 (x − 6) = 1.
12.
Solve (log5 x)2 − 5 log5 x + 4 = 0.
13.
Solve log5 (log3 (log7 x)) = 0.
14.
Solve 2 log2 x = log2 (2x + 15).
15.
Find the value of log3 (1/2) + log3 (2/3) + log3 (3/4) + … + log3 (26/27).
16.
If log10 2 = a and log10 3 = b, express log10 135 in terms of a and b.
Section C — Word problems4 × 4 = 16 marks
17.
(a) On the Richter scale, how many times as strong is an earthquake of magnitude 7 as one of magnitude 3? (b) A solution has [H+] = 10−4 mol per litre. Find its pH, given pH = −log10 [H+]. (c) Another solution is 10 times as acidic. Find its pH.
18.
Solve log10 (x + 8) + log10 (x − 187) = 3, and check each root.
19.
Solve (log5 x)2 + log5 x2 = 24.
20.
Given log10 2 = 0.3010 and log10 3 = 0.4771, find (a) log10 432 (b) log10 2,250 (c) log10 0.72.
AmbifulWorksheets
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CBSE · Class 9 · Mathematics

Answer key

Logarithms (Advanced) · AW-9M-ALO-2026-V1 · For parents — fold back or tear off before handing the sheet to your child.

QAnswerWorking
1(d) 416 = 24
2(c) 4log5 (1375 × 5 ÷ 11) = log5 625 = 4
3(d) 0a0 = 1
4(c) 5change of base: log3 5 × log5 243 = log3 243 = 5
5(d) 2p + 120 = 22 × 5
6(d) 100 times as strongeach unit is 10 times: 102 = 100
7(c) 100[H+] = 10 to the power −pH; 106 ÷ 104 = 102
8(a) x = 35x + 66 = 34 = 81, so 5x = 15
9log2 1123 = log2 8; log2 8 + log2 14 = log2 (8 × 14)
101/225 = 52 and 5 = 51, so the value is 1/2
11x = 7x(x − 6) = 71 = 7; (x − 7)(x + 1) = 0; x = −1 is rejected (log of a negative number)
12x = 5 or x = 625put y = log5 x: (y − 1)(y − 4) = 0, so y = 1 or 4; both values of x are positive
13x = 343log3 (log7 x) = 50 = 1; log7 x = 31 = 3; x = 73
14x = 5x2 = 2x + 15; (x − 5)(x + 3) = 0; x = −3 is rejected (log2 (−3) is not defined)
15−3product rule: log3 (1/2 × 2/3 × … × 26/27) = log3 (1/27) = −3
16−a + 3b + 1135 = 33 × 5, and log10 5 = log10 (10/2) = 1 − a
17(a) 10,000; (b) 4; (c) 3(a) 104; (b) −log10 10−4 = 4; (c) [H+] is 101 times larger, so the pH falls by 1
18x = 192(x + 8)(x − 187) = 1000; x2 − 179x − 2496 = 0; (x − 192)(x + 13) = 0; x = −13 makes both arguments negative, so it is rejected
19x = 625 or x = 1/15625power rule: log5 x2 = 2 log5 x; with y = log5 x: y2 + 2y − 24 = 0, (y − 4)(y + 6) = 0, y = 4 or −6
20(a) 2.6353; (b) 3.3522; (c) −0.1428(a) 4 × 0.3010 + 3 × 0.4771; (b) log10 5 = 1 − 0.3010 = 0.6990; (c) 0.72 = 72 ÷ 102, so subtract 2