9th Class Mathematics - Chapter 2 - Review Exercise Solved

Chapter 2: Logarithms

Review Exercise 2 Solved Notes

A beautifully formatted, step-by-step academic solved guide for textbook Chapter 2 Review Exercise. Features clean equations and interactive logarithm toolboxes, completely watermark-free.

Question 1 Multiple Choice Questions

Choose the correct option for each of the following:

Quick Answer Key

Q	1	2	3	4	5	6	7	8	9	10
Ans	c	b	b	d	a	c	d	c	d	c

i. The standard form of

5.2 \times 10 6

is:

(a) 52,000

(b) 520,000

(d) 52,000,000

Reason: Exponent is positive 6, so shift the decimal point 6 places to the right:

5.2 \times 10 6 = 5,200,000

ii. Scientific notation of

0.00034

is:

(a) 3.4 × 10³

(b) 3.4 × 10^-4

(d) 3.4 × 10^-3

Reason: The first non-zero digit is 3. We shift the decimal point 4 places to the right, which gives a negative exponent:

3.4 \times 10 -4

iii. The base of common logarithm is:

(a) 2

(b) 10

(d) e

Reason: Common logarithms are base 10 logarithms, whereas natural logarithms are base e.

iv.

log 223 = ______

(a) 1

(b) 2

(d) 3

Reason: Applying the Power Law,

log 223 = 3 \times log 2 2 = 3 \times 1 = 3

log 100 = ______

(a) 2

(b) 3

(d) 1

Reason: Common logarithm base is 10. Since

100 = 10 2

log 10102 = 2

vi. If

log 2 = 0.3010

, then

log 200

is:

(a) 1.3010

(b) 0.6010

(d) 2.000

Reason:

log 200 = log (2 \times 100) = log 2 + log 100 = 0.3010 + 2 = 2.3010

vii.

log(0) = ______

(a) positive

(b) negative

(d) undefined

Reason: Logarithm is only defined for strictly positive real numbers (

x > 0

). Thus log(0) is undefined.

viii.

log 10,000 = ______

(a) 2

(b) 3

(d) 5

Reason:

10,000 = 10 4

. Hence,

log 10104 = 4

ix.

log 5 + log 3 = ______

(a) log 0

(b) log 2

(d) log 15

Reason: According to the Product Law,

log m + log n = log (m \times n)

. Thus,

log 5 + log 3 = log(5 \times 3) = log 15

34 = 81

in logarithmic form is:

(a) log₃ 4 = 81

(b) log₄ 3 = 81

(d) log₄ 81 = 3

Reason: Base is 3, power is 4, result is 81. Thus,

log base (result) = power \Rightarrow log 3 81 = 4

Additional Concept MCQs

1. If

a = b \times 10 n

is written in scientific notation, then:

(a) 0 ≤ b ≤ 10

(b) 0 ≤ b < 10

(d) 1 ≤ b < 10

Reason: Scientific notation requires the base coefficient

b

to be greater than or equal to 1 and strictly less than 10.

2. For the value

log 0.00327

, the characteristic is:

(a) -2

(b) -3 (or 3)

(d) 0

Reason: The first non-zero digit (3) is 3 places after the decimal point. Since it is less than 1, characteristic is negative:

-3

(written as bar 3).

3. The characteristic of

log (4.9 \times 10 -5)

is:

(a) -5

(b) 10

(d) 9

Reason: For numbers in scientific notation, the exponent of 10 represents the characteristic directly. Hence, characteristic is -5.

4. If

a x = n

, then:

(a) a = log_x n

(b) x = log_n a

(d) a = log_n x

Reason: By definition of logarithms, exponent

x

is the logarithm of result

n

to the base

a

5. The relation

y = log z x

implies:

(a) x^y = z

(b) z^y = x

(d) y^z = x

Reason: The base of the log becomes the base of the exponent, and the log output becomes the power:

z y = x

6. The logarithm of unity (

1

) to any valid base is:

(a) 1

(b) 10

(d) 0

Reason: Since any non-zero base raised to power 0 equals 1 (

b 0 = 1

log b 1 = 0

7. The logarithm of any number to itself as base is:

(a) 1

(b) 0

(d) 10

Reason: Since

b 1 = b

log b b = 1

8. The base of natural logarithms is:

(a) 0

(b) 1

(d) e

Reason: Natural logarithms (denoted as

ln

) use the Euler's number

e \approx 2.71828

as their base.

9. If

log(x + 3) = log(15x - 4)

, then

x

is:

(a) 0.5

(b) 7

(d) 2

Reason: Equate arguments:

x + 3 = 15x - 4 \Rightarrow 14x = 7 \Rightarrow x = 7/14 = 0.5

10. If

log \sqrtx 25 = 4

, then

x

is:

(a) +5

(b) -5

(d) impossible

Reason: Exponential form:

(\sqrtx) 4 = 25 \Rightarrow x 2 = 25 \Rightarrow x = 5

(base must be positive, so reject -5).

11.

log b g h

is:

(a) g log_b h

(b) log_b(gh)

(d) h log_g b

Reason: By Power Law,

log b g h = h \times log b g

12.

log b x - log b y

is:

(a) log_b x / log_b y

(b) log_b (x / y)

(d) log_b y / log_b x

Reason: The Quotient Law of logarithms states that subtraction of logs equals log of division.

13.

log b a \times log c b

can be written as:

(a) log_c a

(b) log_a c

(d) log_b c / log_b a

Reason: Change of base:

(log a / log b) \times (log b / log c) = log a / log c = log c a

14.

log y x

is equal to:

(a) log_z x / log_y z

(b) log_x z / log_y z

(d) log_z y / log_z x

Reason: According to standard Change of Base Rule, we can choose any new base z.

15.

log (1 / n)

is equal to:

(a) log 1

(b) log n

(d) -log n

Reason:

log (1 / n) = log 1 - log n = 0 - log n = -log n

16. If

log(a / b) + log(b / a) = log(a + b)

, then:

(a) a + b = 1

(b) a - b = 1

(d) a² - b² = 1

Reason:

log(a / b) + log(b / a) = log(a / b \times b / a) = log 1

. Thus,

log 1 = log(a + b) \Rightarrow a + b = 1

17. The common logarithm of e is:

log e = ______

(where

e \approx 2.718

)

(a) 0

(b) 0.4343

(d) 1

Reason:

log 10 (2.71828) \approx 0.434294 \approx 0.4343

18. The natural logarithm of 10 is:

log e 10 = ______

(where

e \approx 2.718

)

(a) 2.3026

(b) 0.4343

(d) 10

Reason:

log e 10 = ln 10 \approx 2.302585 \approx 2.3026

. Note that

ln 10 = 1 / log e

19.

log 9 (1 / 81) = ______

(a) -1

(b) -2

(d) does not exist

Reason:

1 / 81 = 1 / 9 2 = 9 -2

. Hence,

log 99 -2 = -2

20. The simplified value of

log 77 -3 + log 243

is:

(a) 0

(b) -3

(d) ± 3

Reason:

log 77 -3 = -3

and

log 2 (2 2) 3 = log 226 = 6

. Thus

-3 + 6 = 3

21. The value of

log \sqrt10 1002

is:

(a) 2

(b) 1

(d) 8

Reason:

1002 = (10 2) 2 = 10 4 = ( (\sqrt10) 2) 4 = (\sqrt10) 8

. Hence, log value is 8.

22. The value of

log 10100

is:

(a) 2

(b) 0

(d) impossible

Reason:

100 = 1

. Log of 1 to any base is 0.

23. The value of

log 4 + log 25

is:

(a) 2

(b) 3

(d) 5

Reason: Product rule:

log 4 + log 25 = log(4 \times 25) = log 100 = 2

24. Evaluate

log 7 1 \sqrt7

(a) -1

(b) -1/2

(d) 1/7

Reason:

1 / \sqrt7 = 7 -1/2

. Hence, log value is

-1/2

25. If

log b x = 4

and

log b y = 2

, then value of

log b (x \cdot y 3)

is:

(a) 5

(b) 7

(d) 10

Reason: laws:

log b x + 3 log b y = 4 + 3(2) = 4 + 6 = 10

Questions 2 & 3 Notations

Question 2: Express in scientific notation.
Question 3: Express in ordinary notation.

Q2 (i) 0.000567

Solution:

Shift decimal point 4 places to the right to place it after the first non-zero digit 5:

= 5.67 \times 10 -4

Q2 (ii) 734

Solution:

Write decimal point after 3. Shift decimal point 2 places to the left:

= 7.34 \times 10 2

Q2 (iii) 0.33 × 10³

Solution:

1. Convert $0.33$ to standard scientific notation: $0.33 = 3.3 \times 10 -1$ .

2. Multiply by $103$ and apply exponent rules:

= (3.3 \times 10 -1) \times 10 3 = 3.3 \times 10 -1 + 3

= 3.3 \times 10 2

Q3 (i) 2.6 × 10³

Solution:

Since the exponent is positive 3, shift the decimal point 3 places to the right:

= 2600

Q3 (ii) 8.794 × 10^-4

Solution:

Since the exponent is negative 4, shift the decimal point 4 places to the left:

= 0.0008794

Q3 (iii) 6 × 10^-6

Solution:

Since the exponent is negative 6, shift the decimal point 6 places to the left:

= 0.000006

Questions 4 & 5 Format Conversion

Question 4: Express in logarithmic form.
Question 5: Express in exponential form.

Q4 (i) 3⁷ = 2187

Solution:

Recall: $b y = x \Rightarrow log b x = y$ .

log 3 2187 = 7

Q4 (ii) a^b = c

Solution:

log a c = b

Q4 (iii) (12)² = 144

Solution:

log 12 144 = 2

Q5 (i) log₄ 8 = x

Solution:

Recall: $log b x = y \Rightarrow b y = x$ .

4 x = 8

Q5 (ii) log₉ 729 = 3

Solution:

93 = 729

Q5 (iii) log₄ 1024 = 5

Solution:

45 = 1024

Question 6 Solve for x

Find the value of x in the following logarithmic equations:

(i) log₉ x = 0.5

Solution:

1. Convert to exponential form:

9 0.5 = x

2. Since exponent $0.5 = 12$ (square root):

x = \sqrt9

x = 3

(ii) (19)^3x = 27

Solution:

1. Write both bases in powers of 3: $19 = 9 -1 = (3 2) -1 = 3 -2$ , and $27 = 3 3$ .

(3 -2) 3x = 3 3

2. Simplify exponents:

3 -6x = 3 3

3. Equate exponents:

-6x = 3

x = 3 -6

x = - 12 = -0.5

(iii) (132)^2x = 64

Solution:

1. Express both bases in powers of 2: $132 = 32 -1 = (2 5) -1 = 2 -5$ , and $64 = 2 6$ .

(2 -5) 2x = 2 6

2. Simplify exponents:

2 -10x = 2 6

3. Equate exponents:

-10x = 6

x = 6 -10

x = - 35 = -0.6

Question 7 Single Logarithm

Write the following expressions as a single logarithm:

(i) 7 log x - 3 log y²

Solution:

1. Apply the Power Law: $k log m = log (m k)$ .

7 log x = log (x7)   \text{and}   3 log y2 = log ( (y2)3 ) = log (y6)

2. Substitute back and apply the Quotient Law:

log (x 7) - log (y 6)

= log (x 7 y 6)

(ii) 3 log 4 - log 32

Solution:

1. Apply the Power Law:

3 log 4 = log (4 3) = log 64

2. Apply the Quotient Law:

log 64 - log 32 = log (6432)

= log 2

(iii) 13 (log₅ 8 + log₅ 27) - log₅ 3

Solution:

1. Combine sum inside parentheses using the Product Law:

log 5 8 + log 5 27 = log 5 (8 \times 27) = log 5216

2. Apply the Power Law: $13 log 5 216 = log 5 (216 1/3)$ .

Since $216 = 6 3$ , we get $216 1/3 = (6 3) 1/3 = 6$ :

= log 56

3. Substitute back and apply the Quotient Law:

log 5 6 - log 5 3 = log 5 (63)

= log 52

Question 8 Expansion

Expand the following using laws of logarithms:

(i) log (x y z⁶)

Solution:

1. Apply the Product Law:

= log x + log y + log (z 6)

2. Apply the Power Law to the last term:

= log x + log y + 6 log z

(ii) log₃ ⁶√{m⁵ n³}

Solution:

1. Write the root as a fractional exponent:

= log 3 (m 5 n 3) 1/6

2. Apply the Power Law:

= 16 log 3 (m 5 n 3)

3. Apply the Product Law:

= 16 [log 3 (m 5) + log 3 (n 3)]

4. Expand using the Power Law inside brackets:

= 16 [5 log 3 m + 3 log 3 n]

5. Distribute coefficient across terms:

= 56 log 3 m + 12 log 3 n

(iii) log √{8 x³}

Solution:

1. Write factors as standard base-power exponents: $8 = 2 3$ , and write root as power $1/2$ :

= log (2 3 x 3) 1/2 = log ( (2x) 3) 1/2

2. Multiply powers: $3 \times 12 = 32$ .

= log (2x) 3/2

3. Apply the Power Law:

= 32 log (2x)

4. Apply the Product Law:

= 32 (log 2 + log x)   \text{or}   32 log 2 + 32 log x

Question 9 Logarithm Tables

Find the values of the following with the help of logarithm table:

(i) ³√{68.24}

Solution:

1. Let $x = (68.24) 1/3$ . Take log of both sides:

log x = log (68.24) 1/3

2. Apply the Power Law:

log x = 13 log (68.24)

3. Look up log table values: Characteristic of 68.24 is 1. Mantissa under 68 row, 2 col, diff 4 is .8340:

log (68.24) = 1.8340

4. Divide by 3:

log x = 1.8340 3 = 0.6113

5. Take antilog: $x = antilog(0.6113)$ . Characteristic = 0. Mantissa .61 under col 1, diff 3 -> 4083 + 3 = 4086:

x \approx 4.086

(ii) 319.8 × 3.543

Solution:

1. Let $x = 319.8 \times 3.543$ . Take log of both sides:

log x = log (319.8 \times 3.543)

2. Apply the Product Law:

log x = log 319.8 + log 3.543

3. Look up table values:

For 319.8: Characteristic = 2, Mantissa = .5049 ⇒ $log 319.8 = 2.5049$
For 3.543: Characteristic = 0, Mantissa = .5493 ⇒ $log 3.543 = 0.5493$

4. Add the values:

log x = 2.5049 + 0.5493 = 3.0542

5. Take antilog: $x = antilog(3.0542)$ . Characteristic = 3. Mantissa .05 under row 4, diff 2 -> 1132 + 1 = 1133:

x \approx 1133

(iii) 36.12 × 750.9113.2 × 9.98

Solution:

1. Let $x = 36.12 \times 750.9 113.2 \times 9.98$ . Take log on both sides:

log x = log (36.12 \times 750.9 113.2 \times 9.98)

2. Apply logarithm laws to expand:

log x = (log 36.12 + log 750.9) - (log 113.2 + log 9.98)

log x = log 36.12 + log 750.9 - log 113.2 - log 9.98

3. Look up table values:

$log 36.12 = 1.5577$
$log 750.9 = 2.8756$
$log 113.2 = 2.0538$
$log 9.98 = 0.9991$

4. Add and subtract the values:

log x = 1.5577 + 2.8756 - 2.0538 - 0.9991

log x = 4.4333 - 3.0529 = 1.3804

5. Take antilog: $x = antilog(1.3804)$ . Characteristic = 1. Mantissa .38 under col 0, diff 4 -> 2399 + 2 = 2401:

x \approx 24.01

Question 10 Word Problems

In the year 2016, the population of a city was 22 millions and was growing at a rate of 2.5% per year. The function:

p(t) = 22(1.025) t

gives the population in millions, t years after 2016. Use the model to determine in which year the population will reach 35 millions. Round to the nearest year.

Solution:

Given the exponential population model: $p(t) = 22 (1.025) t$ .

1. We want to find the time $t$ when population $p(t) = 35$ million:

35 = 22 (1.025) t

2. Divide both sides by 22:

3522 = (1.025) t

1.5909 \approx (1.025) t

3. Take common logarithm on both sides:

log 1.5909 = log (1.025) t

4. Apply the Power Law to move variable $t$ to the front:

log 1.5909 = t \times log 1.025

5. Substitute log table values ( $log 1.5909 \approx 0.2016$ , $log 1.025 \approx 0.0107$ ):

0.2016 = t \times 0.0107

6. Solve for $t$ :

t = 0.2016 0.0107 \approx 18.84 years

7. Rounding to the nearest year: $t \approx 19$ years.

8. Calculate the calendar year:

2016 + 19 = 2035

Answer: The population will reach 35 millions in the year 2035.

Interactive Concept Sandbox Interactive

Enter a number to convert between ordinary decimal form and scientific notation ( $b \times 10 n$ ).

Decimal Number (e.g. 0.000567, 734):

Verify Change of Base properties or evaluate logarithms of any base: $log base (value)$ .

Base (b):

Value (x):

Practice multiple choice questions testing log laws and conversion properties.

1. If

log 2 = 0.3010

and

log 3 = 0.4771

, what is the value of

log 6

(a) 0.1761

(b) 0.7781 (Correct)

(d) 1.5850

Reason: Product Law:

log 6 = log(2 \times 3) = log 2 + log 3 = 0.3010 + 0.4771 = 0.7781

2. Solve for x:

(&frac19;)3x = 27

(a) x = -1/3

(b) x = -1/2 (Correct)

(d) x = 3/2

Reason:

3 -6x = 3 3 \Rightarrow -6x = 3 \Rightarrow x = -3/6 = -1/2