PrepSureNotes & Solutions

Chapter 2: Logarithms

Exercise 2.2 — Class 9 Mathematics Solutions

All 21 questions below are worked step by step — every line shown, nothing skipped — for the Punjab Board 9th class Mathematics textbook.

Board
Punjab Board (PCTB)
Class
9th Class
Questions solved
21

Exercise 2.2

3 Questions (18 Sub-parts)
Jump to Question
3 Questions • 18 Parts
Q1
(i)(ii)(iii)(iv)(v)(vi)(vii)(viii)
Q2
(i)(ii)(iii)(iv)(v)(vi)(viii)
Q3
(i)(ii)(iii)(iv)(v)(vi)

Q1Part (i)

EasyApproved by Wasif • Aug 23, 2026

103=100010^3 = 1000

Hint

Recall that ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y.

Solution

  1. State the definition of a logarithm.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify the base, exponent, and value.

    • Base: a=10a = 10
    • Exponent: y=3y = 3
    • Value: x=1000x = 1000

  3. Write in logarithmic form.

    log⁡101000=3\log_{10} 1000 = 3

Answer

log⁡101000=3\log_{10} 1000 = 3

Q1Part (ii)

EasyApproved by Wasif • Aug 23, 2026

28=2562^8 = 256

Hint

Base is 22, exponent is 88, and value is 256256.

Solution

  1. State the definition of a logarithm.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify the components.

    • Base a=2a = 2
    • Exponent y=8y = 8
    • Value x=256x = 256

  3. Convert to logarithmic form.

    log⁡2256=8\log_2 256 = 8

Answer

log⁡2256=8\log_2 256 = 8

Q1Part (iii)

EasyApproved by Wasif • Aug 23, 2026

3−3=1273^{-3} = \frac{1}{27}

Hint

Base is 33, exponent is −3-3, and value is 127\frac{1}{27}.

Solution

  1. State the conversion rule.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify the components.

    • Base a=3a = 3
    • Exponent y=−3y = -3
    • Value x=127x = \frac{1}{27}

  3. Write in logarithmic form.

    log⁡3(127)=−3\log_3 \left(\frac{1}{27}\right) = -3

Answer

log⁡3(127)=−3\log_3 \left(\frac{1}{27}\right) = -3

Q1Part (iv)

EasyApproved by Wasif • Aug 23, 2026

202=40020^2 = 400

Hint

Base is 2020, exponent is 22, and value is 400400.

Solution

  1. State the conversion rule.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify the components.

    • Base a=20a = 20
    • Exponent y=2y = 2
    • Value x=400x = 400

  3. Write in logarithmic form.

    log⁡20400=2\log_{20} 400 = 2

Answer

log⁡20400=2\log_{20} 400 = 2

Q1Part (v)

EasyApproved by Wasif • Aug 23, 2026

16−14=1216^{-\frac{1}{4}} = \frac{1}{2}

Hint

Base is 1616, exponent is −14-\frac{1}{4}, and value is 12\frac{1}{2}.

Solution

  1. State the conversion rule.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify the components.

    • Base a=16a = 16
    • Exponent y=−14y = -\frac{1}{4}
    • Value x=12x = \frac{1}{2}

  3. Write in logarithmic form.

    log⁡16(12)=−14\log_{16} \left(\frac{1}{2}\right) = -\frac{1}{4}

Answer

log⁡16(12)=−14\log_{16} \left(\frac{1}{2}\right) = -\frac{1}{4}

Q1Part (vi)

EasyApproved by Wasif • Aug 23, 2026

112=12111^2 = 121

Hint

Base is 1111, exponent is 22, and value is 121121.

Solution

  1. State the conversion rule.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify components.

    • Base a=11a = 11
    • Exponent y=2y = 2
    • Value x=121x = 121

  3. Write in logarithmic form.

    log⁡11121=2\log_{11} 121 = 2

Answer

log⁡11121=2\log_{11} 121 = 2

Q1Part (vii)

EasyApproved by Wasif • Aug 23, 2026

p=qrp = q^r

Hint

Rearrange to qr=pq^r = p. Base is qq, exponent is rr, and value is pp.

Solution

  1. State the conversion rule.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify components.

    • Exponential equation: qr=pq^r = p
    • Base a=qa = q
    • Exponent y=ry = r
    • Value x=px = p

  3. Write in logarithmic form.

    log⁡qp=r\log_q p = r

Answer

log⁡qp=r\log_q p = r

Q1Part (viii)

EasyApproved by Wasif • Aug 23, 2026

(32)−15=12(32)^{-\frac{1}{5}} = \frac{1}{2}

Hint

Base is 3232, exponent is −15-\frac{1}{5}, and value is 12\frac{1}{2}.

Solution

  1. State the conversion rule.

    ay=x  ⟺  log⁡ax=ya^y = x \iff \log_a x = y

  2. Identify components.

    • Base a=32a = 32
    • Exponent y=−15y = -\frac{1}{5}
    • Value x=12x = \frac{1}{2}

  3. Write in logarithmic form.

    log⁡32(12)=−15\log_{32} \left(\frac{1}{2}\right) = -\frac{1}{5}

Answer

log⁡32(12)=−15\log_{32} \left(\frac{1}{2}\right) = -\frac{1}{5}

Q2Part (i)

EasyApproved by Wasif • Aug 23, 2026

log⁡5125=3\log_5 125 = 3

Hint

Use the relationship: log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x.

Solution

  1. State the inverse definition.

    log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x

  2. Identify base, exponent, and value.

    • Base a=5a = 5
    • Exponent y=3y = 3
    • Value x=125x = 125

  3. Write in exponential form.

    53=1255^3 = 125

Answer

53=1255^3 = 125

Q2Part (ii)

EasyApproved by Wasif • Aug 23, 2026

log⁡216=4\log_2 16 = 4

Hint

Base is 22, exponent is 44, and value is 1616.

Solution

  1. State the conversion rule.

    log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x

  2. Identify components.

    • Base a=2a = 2
    • Exponent y=4y = 4
    • Value x=16x = 16

  3. Write in exponential form.

    24=162^4 = 16

Answer

24=162^4 = 16

Q2Part (iii)

EasyApproved by Wasif • Aug 23, 2026

log⁡231=0\log_{23} 1 = 0

Hint

Any non-zero base raised to power 00 equals 11 (a0=1a^0 = 1).

Solution

  1. State the conversion rule.

    log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x

  2. Identify components.

    • Base a=23a = 23
    • Exponent y=0y = 0
    • Value x=1x = 1

  3. Write in exponential form.

    230=123^0 = 1

Answer

230=123^0 = 1

Q2Part (iv)

EasyApproved by Wasif • Aug 23, 2026

log⁡55=1\log_5 5 = 1

Hint

Base is 55, exponent is 11, and value is 55.

Solution

  1. State the conversion rule.

    log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x

  2. Identify components.

    • Base a=5a = 5
    • Exponent y=1y = 1
    • Value x=5x = 5

  3. Write in exponential form.

    51=55^1 = 5

Answer

51=55^1 = 5

Q2Part (v)

EasyApproved by Wasif • Aug 23, 2026

log⁡2(18)=−3\log_2 \left(\frac{1}{8}\right) = -3

Hint

Base is 22, exponent is −3-3, and value is 18\frac{1}{8}.

Solution

  1. State the conversion rule.

    log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x

  2. Identify components.

    • Base a=2a = 2
    • Exponent y=−3y = -3
    • Value x=18x = \frac{1}{8}

  3. Write in exponential form.

    2−3=182^{-3} = \frac{1}{8}

Answer

2−3=182^{-3} = \frac{1}{8}

Q2Part (vi)

EasyApproved by Wasif • Aug 23, 2026

12=log⁡93\frac{1}{2} = \log_9 3

Hint

Base is 99, exponent is 12\frac{1}{2}, and value is 33.

Solution

  1. Rewrite with logarithm on left-hand side.

    log⁡93=12\log_9 3 = \frac{1}{2}

  2. State the conversion rule.

    log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x

  3. Write in exponential form.

    912=39^{\frac{1}{2}} = 3

Answer

912=39^{\frac{1}{2}} = 3

Q2Part (viii)

EasyApproved by Wasif • Aug 23, 2026

log⁡4(116)=−2\log_4 \left(\frac{1}{16}\right) = -2

Hint

Base is 44, exponent is −2-2, and value is 116\frac{1}{16}.

Solution

  1. State the conversion rule.

    log⁡ax=y  ⟺  ay=x\log_a x = y \iff a^y = x

  2. Identify components.

    • Base a=4a = 4
    • Exponent y=−2y = -2
    • Value x=116x = \frac{1}{16}

  3. Write in exponential form.

    4−2=1164^{-2} = \frac{1}{16}

Answer

4−2=1164^{-2} = \frac{1}{16}

Q3Part (i)

EasyApproved by Wasif • Aug 23, 2026

log⁡x64=3\log_x 64 = 3

Hint

Convert to exponential form: x3=64x^3 = 64. Then express 64=4364 = 4^3.

Solution

  1. Convert to exponential form.

    log⁡x64=3x3=64\begin{aligned} \log_x 64 &= 3 \\ x^3 &= 64 \end{aligned}

  2. Express 6464 as a cube power.

    x3=43\begin{aligned} x^3 &= 4^3 \end{aligned}

  3. Equate the bases since powers are equal.

    x=4\begin{aligned} x &= 4 \end{aligned}

Answer

x=4x = 4

Q3Part (ii)

EasyApproved by Wasif • Aug 23, 2026

log⁡51=x\log_5 1 = x

Hint

Convert to exponential form: 5x=15^x = 1. Recall that 50=15^0 = 1.

Solution

  1. Convert to exponential form.

    log⁡51=x5x=1\begin{aligned} \log_5 1 &= x \\ 5^x &= 1 \end{aligned}

  2. Express 11 as a power of base 55.

    5x=50\begin{aligned} 5^x &= 5^0 \end{aligned}

  3. Equate exponents.

    x=0\begin{aligned} x &= 0 \end{aligned}

Answer

x=0x = 0

Q3Part (iii)

EasyApproved by Wasif • Aug 23, 2026

log⁡x8=1\log_x 8 = 1

Hint

Convert to exponential form: x1=8x^1 = 8.

Solution

  1. Convert to exponential form.

    log⁡x8=1x1=8x=8\begin{aligned} \log_x 8 &= 1 \\ x^1 &= 8 \\ x &= 8 \end{aligned}

Answer

x=8x = 8

Q3Part (iv)

MediumApproved by Wasif • Aug 23, 2026

log⁡10x=−3\log_{10} x = -3

Hint

Convert to exponential form: x=10−3=11000x = 10^{-3} = \frac{1}{1000}.

Solution

  1. Convert to exponential form.

    log⁡10x=−3x=10−3\begin{aligned} \log_{10} x &= -3 \\ x &= 10^{-3} \end{aligned}

  2. Simplify negative power.

    x=1103x=11000(or 0.001)\begin{aligned} x &= \frac{1}{10^3} \\ x &= \frac{1}{1000} \quad (\text{or } 0.001) \end{aligned}

Answer

x=11000orx=0.001x = \frac{1}{1000} \quad \text{or} \quad x = 0.001

Q3Part (v)

MediumApproved by Wasif • Aug 23, 2026

log⁡4x=32\log_4 x = \frac{3}{2}

Hint

Convert to exponential form: x=432=(22)32=23=8x = 4^{\frac{3}{2}} = (2^2)^{\frac{3}{2}} = 2^3 = 8.

Solution

  1. Convert to exponential form.

    log⁡4x=32x=432\begin{aligned} \log_4 x &= \frac{3}{2} \\ x &= 4^{\frac{3}{2}} \end{aligned}

  2. Simplify the rational exponent.

    x=(22)32x=22×32x=23x=8\begin{aligned} x &= (2^2)^{\frac{3}{2}} \\ x &= 2^{2 \times \frac{3}{2}} \\ x &= 2^3 \\ x &= 8 \end{aligned}

Answer

x=8x = 8

Q3Part (vi)

MediumApproved by Wasif • Aug 23, 2026

log⁡21024=x\log_2 1024 = x

Hint

Convert to exponential form: 2x=10242^x = 1024. Express 1024=2101024 = 2^{10}.

Solution

  1. Convert to exponential form.

    log⁡21024=x2x=1024\begin{aligned} \log_2 1024 &= x \\ 2^x &= 1024 \end{aligned}

  2. Factorize 10241024 into base 22.

    1024=210\begin{aligned} 1024 &= 2^{10} \end{aligned}

  3. Equate exponents.

    2x=210x=10\begin{aligned} 2^x &= 2^{10} \\ x &= 10 \end{aligned}

Answer

x=10x = 10

The exercises and question numbering reproduced on these pages are from Mathematics for Class 9 (National Curriculum of Pakistan 2023), published by the Punjab Curriculum and Textbook Board (PCTB), Lahore, authored by Muhammad Akhtar Shirani, Madiha Mahmood, and Ghulam Murtaza. PCTB holds the copyright in the original textbook. PrepSure is not affiliated with, endorsed by, or sponsored by PCTB. The worked solutions, explanations, hints and method notes are PrepSure's own original work, written and reviewed by our team. If you hold rights in this material and believe anything here exceeds fair use, write to us and we will take it down.

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