# ICSE Solutions for Selina Concise Chapter 8 Logarithms Class 9 Maths

### Exercise 8(A)

1. Express each of the following in logarithmic form:

(i) 53 = 125

(ii) 3-2 = 1/9

(iii) 10-3 = 0.001

(iv) (81)3/4 = 27

#### 2. Express each of the following in exponential form:

(i) logg 0.125 = -1

(ii) log100.01 = -2

(iii) logaA = x

(iv) log101 = 0

3. Solve for x: log10 x = -2.

4. Find the logarithm of:

(i) 100 to the base 10

(ii) 0.1 to the base 10

(iii) 0.001 to the base 10

(iv) 32 to the base 4

(v) 0.125 to the base 2

(vi) 1/16 to the base 4

(vii) 27 to the base 9

(viii) 1/81 to the base 27

#### 5. State, true or false:

(i) If log10 x = a, then 10x = a.

(ii) If xy = z, then y = logzx.

(iii) log2 8 = 3 and log8 = 2 = 1/3

#### 6. Find x, if:

(i) log3x = 0

(ii) logx2 = -1

(iii) log9243 = x

(iv) log5(x – 7) = 1

(v) log432 = x – 4

(vi) log7(2x2 – 1) = 2

(i) log10 0.01

(ii) log2(1 ÷ 8)

(iii) log51

(iv) log5125

(v) log168

(vi) log0.516

#### 8. If loga m = n, express an–1 in terms in terms of a and m.

9. Given Log2x = m and log5y = n.

(i) Express 2m-3 in terms of x.

(ii) Express 53n+2 in terms of y.

#### 10. If Log2x = a and log5y = a, write 72 in terms of x and y.

11. Solve for x: log(x-1) + log(x+1) = log21.

12. If log (x2 – 21) = 2, show that x = ±11.

### Exercise 8(B)

1. Express in terms of log 2 and log 3:

(i) log 36

(ii) log 144

(iii) log 4.5

(iv) log(26/51)−log(91/119)

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

#### 2. Express each of the following in a form free from logarithm:

(i) 2 log x – log y = 1

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

(iii) a log x – b log y = 2 log 3

#### 3. Evaluate each of the following without using tables:

(i) Log5 +log8−2log2

(ii) Log108 + log1025 + 2log103 – log1018

(iii) Log4 + 1/3(log125) – 1/5(log32)

4. Prove that: 2log(15/18)− log(25/162) + log(4/9) = log2

5. Find x, if: x – log 48 + 3 log 2 = 1/3 log 125 – log 3.

6. Express log102 + 1 in the form of log10x.

7. Solve for x:

(i) log10 (x – 10) = 1

(ii) log (x2 – 21) = 2

(iii) log (x – 2) + log (x + 2) = log 5

(iv) log (x + 5) + log (x – 5) = 4 log 2 + 2 log 3

8. Solve for x:

(i) log81/log27 = x

(ii) log128/log32 = x

(iii) log64/log8 = logx

(iv) log225/log15 = logx

9. Given log x = m + n and log y = m – n, express the value of log ….. in terms of m and n.

10. State, true or false:

(i) log 1 log 1000 = 0

(ii) logx/logy = logx −logy

(iii) If then x = 2. Find log25/log5 = logx

(iv) log x log y = log x + log y

11. If log102 = a and log103 = b; express each of the following in terms of ‘a’ and ‘b’:

(i) log 12

(ii) log 2.25

(iii) log 2(1/4)

(iv) log 5.4

(v) log 60

(iv) log 3(1/8)

12. If log 2 = 0.3010 and log 3 = 0.4771; find the value of:

(i) log 12

(ii) log 1.2

(iii) log 3.6

(iv) log 15

(v) log 25

(vi) 2/3 log 8

13. Given 2 log10 x + 1 = log10 250, find :

(i) x

(ii) log10 2x

14. Given 3logx + (1/2) logy = 2, express y in terms of x  .

15. If X= (100)a, y = (10000)b and z = (10)c , find log(10√y/x2z3) in terms of a, b and c.

16. If 3(log5−log3)−(log5−2log6) = 2−logx, find x.

### Exercise 8(C)

1. If log10 8 = 0.90; find the value of:

(i) log104

(ii) log √32

(iii) log 0.125

2. If log 27 = 1.431, find the value of :

(i) log 9

(ii) log 300

3. If log10 a = b, find 103b – 2 in terms of a.

4. If log5 x = y, find 52y+ 3 in terms of x.

5. Given: log3 m = x and log3 n = y.

(i) Express 32x – 3 in terms of m.

(ii) Write down 31 – 2y + 3x in terms of m and n.

(iii) If 2 log3 A = 5x – 3y; find A in terms of m and n.

6. Simplify:

(i) log (a)3 – log a

(ii) log (a)3 +log a

7. If log (a + b) = log a + log b, find a in terms of b.

8. Prove that:

(i) (log a)2 – (log b)2 = log a/b . log (ab)

(ii) If a log b + b log a – 1 = 0, then ba.ab = 10

9. (i) If log (a + 1) = log (4a – 3) – log 3; find a.

(ii) If 2 log y – log x – 3 = 0, express x in terms of y.

(iii) Prove that: log10 125 = 3(1 – log102).

10. Give log x = 2m –n , log y = n –2n and log z = 3m –2n , find in terms of m and n, the value of : log(x2y3/z4)

11. Give logx25 − logx5=2−logx(1/125); find x .

### Exercise 8(D)

1. If 3/2 log a + 2/3 log b – 1 = 0, find the value of a9.b4.

2. If x = 1 + log 2 – log 5, y = 2 log3 and z = log a – log 5; find the value of a if x + y = 2z.

3. If x = log 0.6; y = log 1.25 and z = log 3 – 2 log 2, find the values of:

(i) x+y- z

(ii) 5x + y – z

4. If a2 = log x, b3 = log y and 3a2 – 2b3 = 6 log z, express y in terms of x and z.

5. If log(a –b )/2 = 1/2(log a + log b), show that: a+ b2 = 6ab.

6. If a2 + b2 = 23ab, show that:

log[(a+b)/5] = 1/2(log a + log b).

7. If m = log 20 and n = log 25, find the value of x, so that: 2 log (x – 4) = 2 m – n.

8. Solve for x and y ; if x > 0 and y > 0;log xy = log x/y + 2 log 2 = 2.

#### 9. Find x, if:

(i) logx 625 = -4

(ii) logx (5x – 6) = 2

10. If p =log 20 and q = log 25, find the value of x, if 2 log(x+1) = 2p – q .

11. If Log2(x+y) = log3(x –y ) = log25/log 0.2 , find the values of x and y.

12. Given : log x/log y = 3/2 and log(xy) = 5; find the values of x and y.

#### 13. Given log10x = 2a and log10y = b/2

(i) Write 10a in terms of x.

(ii) Write 102b + 1 in terms of y.

(iii) If log10P = 3a –2b,  express P in terms of x and y.

14. Solve: log5(x + 1) – 1 = 1 + log5(x – 1).

15. Solve for x, if: log, 49 – log x 7+ log x (1/343) = −2

16. if a2 = log x, b3 = log y and a2/2 –b3/3 = log c , find c in terms of x and y.

#### 17. Give x=log 10 12, y = log42x log10 9 and z = log10 0.4, find :

(i) x – y – z

(ii) 13x – y – z

18. Solve for x,

logx 15√5 = 2 –logx 3√5

19. Evaluate

(i)logba × logcb ×logac

(ii)log38 ÷log9 16

(iii)log 58/(log25 16 × log 100 10)

20. Show that: logam ÷ log ab