## ML Aggarwal Class 9 Solutions for ICSE Maths Chapter 9 Logarithms

**Exercise 9.1**

**Question 1.**

**Convert the following to logarithmic form:**

**(i) 5 ^{2} = 25**

**(ii) a**

^{5}=64**(iii) 7**

^{x}=100**(iv) 9° = 1**

**(v) 6**

^{1}= 6**(vi) 3**

^{-2}= \(\frac { 1 }{ 9 }\)**(vii) 10**

^{-2}= 0.01**(viii) (81)**

^{\(\frac { 3 }{ 4 }\)}= 27**Solution:**

**Question 2.**

**Convert the following into exponential form:**

**(i) log _{2} 32 = 5**

**(ii) log**

_{3}81=4**(iii) log**

_{3}\(\frac { 1 }{ 3 }\)= -1**(iv) log**

_{3}4= \(\frac { 2 }{ 3 }\)**(v) log**

_{8}32= \(\frac { 5 }{ 3 }\)**(vi) log**

_{10}(0.001) = -3**(Vii) log**

_{2}0.25 = -2**(viii) log**

_{a}(\(\frac { 1 }{ a }\)) =-1**Solution:**

**Question 3.**

**By converting to exponential form, find the values of:**

**(i) log _{2} 16**

**(ii) log**

_{5}125**(iii) log**

_{4}8**(iv) log**

_{9}27**(v) log**

_{10}(.01)**(vi) log**

_{7}\(\frac { 1 }{ 7 }\)**(vii) log**

_{5}256**(Viii) log**

_{2}0.25**Solution:**

**Question 4.**

**Solve the following equations for x.**

**Solution:**

**Question 5.**

**Given log _{10}a = b, express 10^{2b-3} in terms of a.**

**Solution:**

**Question 6.**

**Given log _{10} x= a, log_{10} y = b and log_{10} z =c,**

**(i) write down 10**

^{2a-3}in terms of x.**(ii) write down 10**

^{3b-1}in terms of y.**(iii) if log**

_{10}P = 2a + \(\frac { b }{ 2 }\)– 3c, express P in terms of x, y and z.**Solution:**

**Question 7.**

**If log _{10}x = a and log_{10}y = b, find the value of xy.**

**Solution:**

**Question 8.**

**Given log _{10} a = m and log_{10} b = n, express \(\frac { { a }^{ 3 } }{ { b }^{ 2 } }\) in terms of m and n.**

**Solution:**

**Question 9.**

**Given log _{10}a= 2a and log_{10}y = –\(\frac { b }{ 2 }\)**

**(i) write 10**

^{a}in terms of x.**(ii) write 10**

^{2b+1}in terms of y.**(iii) if log**

_{10}P= 3a -2b, express P in terms of x and y .**Solution:**

**Question 10.**

**If log _{2} y = x and log_{3} z = x, find 72^{x} in terms of y and z.**

**Solution:**

**Question 11.**

**If log _{2} x = a and log_{5}y = a, write 100^{2a-1} in terms of x and y.**

**Solution:**

**Exercise 9.2**

**Question 1.**

**Simplify the following :**

**Solution:**

**Question 2.**

**Evaluate the following:**

**Solution:**

**Question 3.**

**Express each of the following as a single logarithm:**

**Solution:**

**Question 4.**

**Prove the following :**

**(i) log _{10} 4 ÷ log_{10} 2 = l0g_{3} 9**

**(ii) log**

_{10}25 + log_{10}4 = log_{5}25**Solution:**

**Question 5.**

**If x = 100) ^{a} , y = (10000)^{b} and z = (10)^{c}, express**

**Solution:**

**Question 6.**

**If a = log _{10}x, find the following in terms of a :**

**(i) x**

**(ii) log**

_{10}\(\sqrt [ 5 ]{ { x }^{ 2 } }\)**(iii) log**

_{10}5x**Solution:**

**Question 7.**

**Solution:**

**Question 8.**

**Solution:**

**Question 9.**

**If x = log _{10} 12, y = log_{4} 2 x log_{10} 9 and z = log_{10} 0.4, find the values of**

**(i)x-y-z**

**(ii) 7**

^{x-y-z}**Solution:**

**Question 10.**

**If log V + log3 = log π + log4 + 3 log r, find V in terns of other quantities.**

**Solution:**

**Question 11.**

**Given 3 (log 5 – log3) – (log 5-2 log 6) = 2 – log n , find n.**

**Solution:**

**Question 12.**

**Given that log _{10}y + 2 log_{10}x= 2, express y in terms of x.**

**Solution:**

**Question 13.**

**Express log _{10}2+1 in the from log_{10}x.**

**Solution:**

**Question 14.**

**Solution:**

**Question 15.**

**Given that log m = x + y and log n = x-y, express the value of log m²n in terms of x and y.**

**Solution:**

**Question 16.**

**Solution:**

**Question 17.**

**Solution:**

**Question 18.**

**Solve for x:**

**Solution:**

**Question 19.**

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

**(i) x**

**(ii) log**

_{10}2x**Solution:**

**Question 20.**

**Solution:**

**Question 21.**

**Prove the following :**

**(i) 3 ^{log 4} = 4^{log 3}**

(ii) 27^{log 2} = 8^{log 3}

**Solution:**

**Question 22.**

**Solve the following equations :**

**(i) log (2x + 3) = log 7**

**(ii) log (x +1) + log (x – 1) = log 24**

**(iii) log (10x + 5) – log (x – 4) = 2**

**(iv) log _{10}5 + log_{10}(5x+1) = log_{10}(x + 5) + 1**

**(v) log (4y – 3) = log (2y + 1) – log3**

**(vi) log**

_{10}(x + 2) + log_{10}(x – 2) = log_{10}3 + 31og_{10}4.**(vii) log(3x + 2) + log(3x – 2) = 5 log 2.**

**Solution:**

**Question 23.**

**Solve for x :**

**log _{3} (x + 1) – 1 = 3 + log_{3} (x – 1)**

**Solution:**

**Question 24.**

**Solution:**

**Question 25.**

**Solution:**

**Question 26.**

**Solution:**

**Question 27.**

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

**Solution:**

**Question 28.**

**Show that:**

**Solution:**

**Question 29.**

**Prove the following identities:**

**Solution:**

**Question 30.**

**Solution:**

**Question 31.**

**Solve for x :**

**Solution:**

**Multiple Choice Questions**

**correct Solution from the given four options (1 to 7):**

**Question 1.**

**If log√3 27 = x, then the value of x is**

**(a) 3**

**(b) 4**

**(c) 6**

**(d) 9**

**Solution:**

**Question 2.**

**If log _{5} (0.04) = x, then the vlaue of x is**

**(a) 2**

**(b) 4**

**(c) -4**

**(d) -2**

**Solution:**

**Question 3.**

**If log _{0.5} 64 = x, then the value of x is**

**(a) -4**

**(b) -6**

**(c) 4**

**(d) 6**

**Solution:**

**Question 4.**

**If log _{10\(\sqrt [ 3 ]{ 5 }\)} x = -3, then the value of x is**

**Solution:**

**Question 5.**

**If log (3x + 1) = 2, then the value of x is**

**Solution:**

**Question 6.**

**The value of 2 + log _{10} (0.01) is**

**(a)4**

**(b)3**

**(c)1**

**(d)0**

**Solution:**

**Question 7.**

**Solution:**

**Chapter Test**

**Question 1.**

**Solution:**

**Question 2.**

**Find the value of log√3 3√3 – log _{5} (0.04)**

**Solution:**

**Question 3.**

**Prove the following:**

**Solution:**

**Question 4.**

**If log (m + n) = log m + log n, show that n = \(\frac { m }{ m-1 }\)**

**Solution:**

**Question 5.**

**Solution:**

**Question 6.**

**Solution:**

**Question 7.**

**Solve the following equations for x:**

**Solution:**

**Question 8.**

**Solve for x and y:**

**Solution:**

**Question 9.**

**If a = 1 + log _{x}yz, 6 = 1+ log_{y} zx and c=1 + log_{z}xy, then show that ab + bc + ca = abc.**

**Solution:**