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
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2. Express each of the following in exponential form:
(i) logg 0.125 = -1
(ii) log100.01 = -2
(iii) logaA = x
(iv) log101 = 0
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3. Solve for x: log10 x = -2.
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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
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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
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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
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7. Evaluate:
(i) log10 0.01
(ii) log2(1 ÷ 8)
(iii) log51
(iv) log5125
(v) log168
(vi) log0.516
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8. If loga m = n, express an–1 in terms in terms of a and m.
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9. Given Log2x = m and log5y = n.
(i) Express 2m-3 in terms of x.
(ii) Express 53n+2 in terms of y.
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10. If Log2x = a and log5y = a, write 72 in terms of x and y.
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11. Solve for x: log(x-1) + log(x+1) = log21.
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12. If log (x2 – 21) = 2, show that x = ±11.
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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)
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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
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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)
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4. Prove that: 2log(15/18)− log(25/162) + log(4/9) = log2
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5. Find x, if: x – log 48 + 3 log 2 = 1/3 log 125 – log 3.
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6. Express log102 + 1 in the form of log10x.
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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
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8. Solve for x:
(i) log81/log27 = x
(ii) log128/log32 = x
(iii) log64/log8 = logx
(iv) log225/log15 = logx
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9. Given log x = m + n and log y = m – n, express the value of log ….. in terms of m and n.
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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
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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)
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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
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13. Given 2 log10 x + 1 = log10 250, find :
(i) x
(ii) log10 2x
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14. Given 3logx + (1/2) logy = 2, express y in terms of x .
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15. If X= (100)a, y = (10000)b and z = (10)c , find log(10√y/x2z3) in terms of a, b and c.
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16. If 3(log5−log3)−(log5−2log6) = 2−logx, find x.
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Exercise 8(C)
1. If log10 8 = 0.90; find the value of:
(i) log104
(ii) log √32
(iii) log 0.125
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2. If log 27 = 1.431, find the value of :
(i) log 9
(ii) log 300
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3. If log10 a = b, find 103b – 2 in terms of a.
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4. If log5 x = y, find 52y+ 3 in terms of x.
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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.
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6. Simplify:
(i) log (a)3 – log a
(ii) log (a)3 +log a
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7. If log (a + b) = log a + log b, find a in terms of b.
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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
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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).
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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)
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11. Give logx25 − logx5=2−logx(1/125); find x .
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Exercise 8(D)
1. If 3/2 log a + 2/3 log b – 1 = 0, find the value of a9.b4.
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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.
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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
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4. If a2 = log x, b3 = log y and 3a2 – 2b3 = 6 log z, express y in terms of x and z.
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5. If log(a –b )/2 = 1/2(log a + log b), show that: a2 + b2 = 6ab.
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6. If a2 + b2 = 23ab, show that:
log[(a+b)/5] = 1/2(log a + log b).
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7. If m = log 20 and n = log 25, find the value of x, so that: 2 log (x – 4) = 2 m – n.
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8. Solve for x and y ; if x > 0 and y > 0;log xy = log x/y + 2 log 2 = 2.
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9. Find x, if:
(i) logx 625 = -4
(ii) logx (5x – 6) = 2
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10. If p =log 20 and q = log 25, find the value of x, if 2 log(x+1) = 2p – q .
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11. If Log2(x+y) = log3(x –y ) = log25/log 0.2 , find the values of x and y.
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12. Given : log x/log y = 3/2 and log(xy) = 5; find the values of x and y.
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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.
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14. Solve: log5(x + 1) – 1 = 1 + log5(x – 1).
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15. Solve for x, if: log, 49 – log x 7+ log x (1/343) = −2
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16. if a2 = log x, b3 = log y and a2/2 –b3/3 = log c , find c in terms of x and y.
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17. Give x=log 10 12, y = log42x log10 9 and z = log10 0.4, find :
(i) x – y – z
(ii) 13x – y – z
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18. Solve for x,
logx 15√5 = 2 –logx 3√5
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19. Evaluate
(i)logba × logcb ×logac
(ii)log38 ÷log9 16
(iii)log 58/(log25 16 × log 100 10)
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20. Show that: logam ÷ log ab m
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21. If log√27 x = 2(2/3),find x .
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22. Evaluate: 1/(logabc + 1) + 1/(logbca + 1) + 1/(logcab +1)
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