**Selina Concise Class 9 Maths Chapter 8 ****Logarithms**** Exercise 8B Solutions**

**EXERCISE – 8B**

**(1) Express in terms of log2 and log 3 :**

**(i) log 36**

Solution :

log 36

log (2×2×3×3)

log (2^{2} ×3^{2})

log2^{2} + log3^{2} {∵ log (a×b) = log a + log b}

2 log2 + 3 log3

**(ii) log 144**

Solution :

log (2×2×2×2×3×3)

log (2^{4} ×3^{2})

log 2^{4} + log 3^{2}

4 log2 + 2 log3

**(iii) log 4.5**

Solution :

log 4.5

log (45/10)

log (9/2)

log 9 – log 2 {log (a/b) = log a – log b}

log 3^{2} – log 2

2 log 3 – log 2

**(iv) log 26/51 – log 91/119**

Solution :

log 26/51 – log 91/119

= log (26/51/91/119)

= log 26/51 × 119/91

= log (2/3)

= log2 – log3

**(v) log 75/16 – 2log 5/9 + log 32/243**

Solution :

log 75/16 – 2 log 5/9 + log 32/243

= log (3×5×5/2^{4}) – 2 log5/9 + log 2^{5}/3^{5}

= log3 + log5^{2} – log2^{4} – 2 {log5 – log3^{2}) + log2^{5} – log 3^{5}

= log3 + 2log5 – 4log2 – 2log5 + 4log3 + 5log2 – 5log 3

= 5log3 – 4log2 + 5log2 – 5log3

= – 4log2 + 5log2

= log2

**(2) Express each of the following in a form free from logarithm:**

**(i) 2 log x – log y = 1**

Solution :

2 log x – log y = 1

log x^{2} – logy = 1

log_{10} (x^{2}/y) = 1

(10)^{1} = x^{2}/y

10 = x^{2}

10y = x^{2}

**(ii) 2 log x + 3 log y = log a**

Solution :

2 log x + 3 log y = log a

=> log x^{2} + log y^{3} = log a

=> log (x^{2} × y^{3}) = log a

=> x^{2}y^{3} = a

**(iii) a log x – b log y = 2 log 3**

Solution :

a log x – b log y = 2 log 3

=> log x^{a} – log y^{b} = log 3^{2}

=> log x^{a}/y^{b} = log 9

=> x^{a}/y^{b} = 9

=> x^{a} = 9y^{b}

**(3) Evaluate each of the following without using tables :**

**(i) log 5 + log 8 – 2 log 2**

Solution :

log 5 + log 8 – 2 log 2

=> log (5×8) – 2 log 2

=> log 40 – log 2^{2}

=> log 40 – log 4

=> log (40/4)

=> log 10

=> 1

**(ii) Log _{10 }8 + log_{10} 25 + 2 log_{10} 3 – log_{10} 18**

Solution :

log_{10} 8 + log_{10} 25 + 2 log_{10} 3 – log_{10} 18

= log_{10} (8×25) + log 3^{2} – log_{10} 18

= log_{10} (8×25×3^{2}) – log_{10} 18

= log_{10} (8×25×9) – log_{10} 18

= log_{10} (1800) – log_{10} 18

= log_{10} (1800/18)

= log_{10} (100)

= log_{10} 10^{2}

= 2 log_{10} 10

= 2 × 1

= 2

**(iii) log 4 + 1/3 log 125 – 1/5 log 32**

Solution :

log4 + 1/3 log 125 – 1/5 log 32

= log4 + log (125)^{1/3} – log (32)^{1/5}

= log4 + log5 – log2

= log (4×5) – log2

= log (4×5/2)

= log (20/2)

= log 10

= 1

**(4) Prove that :**

**2 log 15/18 – log 25/162 + log 4/9 = log2**

**Solution :**

2 log 15/18 – log 25/162 + log 4/9 = log2

L.H.S. = 2 log 15/18 – log 25/162 + log 4/9

=> log (15/18)^{2} – log 25/162 + log 4/9

=> log 225/324 – log 25/162 + log 4/9

=> log (225/324 × 4/9 / 25/162)

=> log (25/21 × 162/25)

=> log 2

L.H.S. = R.H.S.

**(5) Find x, if :**

**x – log48 + 3 log2 = 1/3 log 125 – log3**

Solution :

x – log48 + 3 log2 = 1/3 log125 – log3

x – log48 + log2^{3} = log (125)^{1/3 }– log3

x = log48 – log8 + log5 – log3

x = log (48×5/8×3)

x = log10

x = 10 (∵ log_{10} 10 = 1)

**(6) Express log _{10} 2 + 1 in the form of log_{10 }x.**

Solution :

log

_{10}2 + 1

= log

_{10}2 + log

_{10}10 (∵ log

_{10}10 = 1)

= log

_{10 }(2 × 10)

= log

_{10}20

**(7) Solve for x :**

**(i) log _{10} (x – 10) = 1**

Solution :

log

_{10}(x – 10) = 1

10

^{1}= x – 10

– x = – 10 -10

– x = – 20

x = 20

**(ii) log (x ^{2} – 21) = 2**

Solution :

log_{10} (x^{2} – 21) = 2

10^{2} = x^{2} – 21

100 = x^{2} – 21

100 + 21 = x^{2}

121 = x^{2}

Taking square root on both sides,

√121 = √x^{2}

± 11 = x

**(iii) log(x – 2) + log(x + 2) = log5**

Solution :

log(x – 2) + log(x + 2) = log5

log (x – 2) (x + 2) = log5 (∵ log a + log b = log ab)

log ((x^{2} – 4) = log5

x^{2} – 4 = 5

x^{2} = 5 + 4

x^{2} = 9

Taking square root on both sides,

√x^{2} = √9

x = ± 3

**(iv) log (x + 5) + log (x – 5) = 4 log2 + 2 log3**

Solution :

log(x + 5) + log(x – 5) = 4 log2 + 2 log3

= log [(x + 5) (x – 5)] = log2^{4} + log3^{2}

= log [x^{2} – 25] = log16 + log9

= log (x^{2} – 25) = log (16 × 9)

= log (x^{2} – 25) = log (144)

= x^{2} – 25 = 144

= x^{2} = 144 + 25

= x^{2} = 169

Taking square root on both sides,

√x^{2} = √169

x = ± 13

**(8) Solve for x :**

**(i) log 81/log 27 = x**

Solution :

log 81/log 27 = x

log 3^{4}/log 3^{3} = x

4 log3/3 log3

4/3 = x

**(ii) log 128/log 32 = x**

Solution :

log 128/log 32 = x

log 2^{7}/log 2^{5} = x

7 log2/5 log2 = x

7/5 = x

**(iii) log 64/log 8 = log x**

Solution :

log 64/log 8 = log x

log 2^{6}/ log 2^{3} = log x

6 log2/3 log3 = log x

6/3 = log x

2 = log x

log x = 2

log_{10} x = 2

10^{2} = x

100 = x

**(iv) log 225/log 15 = log x**

Solution :

log 225/log 15 = log x

=> log 15^{2}/log 15 = log x

=> 2 log15/log15 = log x

=> 2 = log x

=> log x = 2

=> log_{10} x = 2

=> 10^{2} = x

=> 100 = x

**(9) Given log x = m + n and log y = m – n, express the value of log 10x/y ^{2} in terms of m and n.**

Solution :

log 10x/y

^{2}

log10 + log x – log y

^{2}

1 + m + n – 2 log y (∵ log 10 = 1)

1 + m + n – 2 (m – n)

1 + m + n – 2m – 2n

1 – m + 3n

**(10) If log _{10} 2 = a and log_{10 }3 = b; express each of the following in terms of ‘a’ and ‘b’.**

**(i)**

**log 12**

Solution :

log

_{10}2 = a and log

_{10}3 = b

log

_{12}= log (2 × 2 × 3)

= log (2

^{2}× 3)

= log2

^{2}+ log3

= 2 log2 + log3

= 2a + b (∵ Given)

**(ii) log 2.25**

Solution :

log 2.25

= log 225/100

= log (9/4)

= log 9 – log 4

= log 3^{2} – log 2^{2}

= 2 log3 – 2 log2

= 2b – 2a (∵ Given)

**(iii) log 2 1/4**

Solution :

log 2 1/4

= log 8+1/4

= log (9/4)

= log9 – log4

= log3^{2} – log2^{2}

= 2 log3 – 2 log2

= 2b – 2a (∵ Given)

**(iv) log 5.4**

Solution :

log 5.4

= log 54/10

= log 54 – log 10

= log 54 – 1

= log (3×3×3×2) – 1

= log3^{3} + log2 – 1

= 3 log3 + log2 – 1

= 3b + a – 1

**(v) Log 3 1/8**

Solution :

log 3 1/8

= log 24 + 1/8

= log 25/8

= log 25 – log 8

= log 5^{2} – log 2^{3}

= 2 log5 – 3 log2

= 2 log (10/2) – 3 log2

= 2 (log10 – log2) – 3 log2

= 2 log10 – 2 log2 – 3 log2

= 2 × 1 – 2a – 3b

= 2 – 2a – 3b (Given)

**(vi)** **log60**

Solution :

log 60

= log (2×2×3×5)

= log (2×3×10)

= log2 + log3 + log10

= a + b + 1 (∵ Given)

**(12) If log2 = 0.3010 and log3 = 0.4771 ; find the value of :**

**(i)** **log 12**

Solution :

log 12 = log (2×2×3)

= log (2^{2} × 3)

= log2^{2} + log3

= 2 log2 + log3

= 2 (0.3010) + 0.4771

= 0.6020 + 0.4771

= 1.0791

(**ii**) **log 1.2**

Solution :

log 1.2 = log 12/10

= log (2×2×3)/10

= log (2^{2} × 3)/10

= log2^{2} + log3 – log10

= 2 log2 + log3 – log10

= 2 × 0.3010 + 0.4771 – 1

= 1.0791 – 1

= 0.0791

(iii) **log 3.6**

Solution :

log 3.6

= log 36/10

= log (2×2×3×3)/10

= log2^{2} + log3^{2} – log10

= 2 log2 + 2 log3 – 1

= 2 × (0.3010) + 2 × (0.4771) – 1

= 0.6020 + 0.9542 – 1

= 1.5562 – 1

= 0.5562

**(iv) log 15**

Solution :

log 15

= log (15 × 2)/2

= log (30/2)

= log (3×10/2)

= log3 + log10 – log2

= 0.4771 + 1 – 0.3010

= 1.1761

**(v) log 25**

Solution :

log 25 dividing and multiplying by 4.

= log 25 × 4/4

= log 100/4

= log 10^{2}/4

= log 10^{2}/2^{2}

= log 10^{2} – log 2^{2}

= 2 log 10 – 2 log 2

= 2 × 1 – 2 × (0.3010)

= 2 – 0.6020

= 1.398

**(vi) 2/3 log 8
**Solution :

2/3 log 8

= 2/3 log 2

^{3}

= 2/3 × 3 log2

= 2 log2

= 2 × (0.3010)

= 0.6020

**(13) Given 2 log _{10 }x + 1 = log_{10 }250, find :**

(i) x

Solution :

Given that : 2 log_{10} x + 1 = log_{10} 250

log_{10} x^{2} + log_{10} 10 = log_{10 }250 {∵ 1 = log_{10}}

log_{10} (x^{2} × 10) = log_{10} 250

10x^{2} = 250

x^{2} = 250/10

x^{2} = 25

Taking square root on both sides,

√x^{2} = √25

x = ± 5.

**(14) Given 3 log x + 1/2 log y = 2, express y in term of x.**

Solution :

3 log x + 1/2 log y = 2

log_{10 }x^{3} + log y^{1/2 }= 2

log_{10} x^{3} + log √y = 2

log_{10} x^{2} √y= 2

10^{2} = x^{3} √y

100 = x^{3} √y

Squaring on both sides,

(100)^{2} = (x^{3})^{2} (√y)^{2}

10000 = x^{6} y

10000/x^{6} = y

1000x^{-6} = y

**(15) If x = (100) ^{a}, y = (10000)^{b} and z = (10)^{c}, find log 10√y/x^{2} z^{3} in terms of a, b and c.**

Solution :

log 10√y/x

^{2}z

^{3}

= log10 √(10000)

^{b}/((100)

^{a})

^{2}((10)

^{c})

^{3}

= log10 × √(10

^{4b})/(10

^{2})

^{2a}(10

^{3c})

= log10 × √(10)

^{4b}/(10)

^{4a}× (10)

^{3c}

= log10 + log (10

^{4b})

^{1/2 }– log10

^{4a }– log

^{3c}

= 1 + log10

^{2b}– 4a log10 – 3c log10

= 1 + 2b log10 – 4a × 1 – 3c × 1

= 1 + 2b × 1 – 4a – 3c

= 1 + 2b – 4a – 3c

**(16) If 3 (log5 – log3) – (log5 – 2log6) = 2 – log x, find x.**

Solution :

3 log5 – 3 log3 – log5 + 2 log(2×3) – 2 – log x.

2 log5 – 3 log3 + 2 log2 + 2 log3 – 2 – log x.

log 5^{2 }– log 3 + 2 log2 = 2 – log x

log 25 – log 3 + log 2^{2} = 2 – log x

log 25 – log 3 + log 4 = 2 – log x

log (25 × 4/3) = 2 – log x

log 100 – log 3 = 2 – log x

log 10^{2} – log3 = 2 – log x

2 log 10 – log3 = 2 – log x

2 × 1 – log3 = 2 – log x

2 – log 3 = 2 – log x

– log3 = – log x

log3 = log x

log x = log3

x = 3

**Here is your solution of Selina Concise Class 9 Maths Chapter 8 Logarithms Exercise 8B**

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