Questions on Logarithmic Functions and Indices

1.            Solve the following equations:

a)      i) 3^2x = 1000             (Cambridge IGCSE A-Maths 0606/23, May/June 2012 P2 Q5)

      ii) 36^2y-5/6^3y = 6^2y-1/216^y+6 

b)      4^x / 2^5-x = 2^4x / 8^x-3   (IGCSE Q)

c)      3^x+1 + 3^2-x = 28    (IGCSE Q)

d)      2^3x = 8 + 2^{3x – 1}   (SPM 2011 P1 Q7 @ pg 162)

e)      3^{x + 2} – 3^x = 8/9    (SPM 2010 P1 Q7 @ pg 136)

f)        16^2x-3 = 8^4x            (SPM 2008 P1 Q7 @ pg 83)

g)      9(3^{n – 1}) = 27ⁿ      (SPM 2007 P1 Q8 @ pg 58)

h)      3^{n – 3} x 27ⁿ =243   (SPM 2009 P1 Q7 @ pg 110)

i)        2^{x + 4} – 2^{x + 3} = 1   (SPM 2005 P1 Q7 @ pg 5)

j)        8^2x-3 = 1/√(4^{x + 2})  (SPM 2006 P1 Q6 @ pg 31)

k)      27(3^{2x + 4}) = 1     (SPM 2012 P1 Q7 @ pg 187)

2.      (i)  Write 125 = 5³ as a logarithmic equation

(ii) Express log_b32 = 5 in index form

 (iii) Evaluate:

(a)      b^log_b^x 

(b)     10^lg10

(c)      e^{ln x} for x = 3                                   

3.      For each of the following equations, sketch on separate axes its graph, showing clearly where the graph crosses the axis:

a)      y = e^x                                                                           (2)

b)      y = log₃ x                                                                      (2)

c)      y = 2^-x                                                                          (2)

d)      y = log₃ (-x)                                                                  (2)

                                      

4.      Solve the equations:

a)      log_x 128 = 7                                                                 (2)

b)      log₅ (7x -1) = 3                                                            (3)

c)      log₄ t = 6 log_t 16 -1                                                      (5)

5.      Solve

(a)   log_q 343 = 3                                                                             (2)

(b)   log₄ (5n + 9) = 3                                                                       (3)

(c)   log_m 4 + 8 log₄ m = 6                                                                (6)

(d)   2 log₃ x – 3x log₃ x + 6x = 4                                                     (5)

6.      Solve

(a) log_x 125 = 3                                                                             (2)

(b) log₄ (9y + 4) = 4                                                                       (3)

(c) 3 – log_p p = log_p 9                                                                     (6)

7.      Given that f(x) = log₅ 3 + log₅ 6 + log₅ 9 + log₅ 12 + log₅ 15

a)      Show that f(x) = 6 log₅ 3 + 3 log₅ 2 + 1                        (3)

b)      Solve f(x) = 1 + log₅ x + log₅ x²                                    (3)

8.      a) Given that log₈ 7 = m log₂ 7, find the value of m.                          (2)

      b) f(x) = 16x log₉ 10 – 12 log₉ 10 + 4x log₃ x – 3 log₃ x

i)        Factorise f(x) completely                                                     (3)

ii)   Hence, solve f(x) = 0                                                          (3)

9.       Solve the simultaneous equations:                                                  

(a)    2 log₃ x + 3 log₅ y = 7

                                 log₃ x – log₅ y = 1                                                (4)

(b)   (Given that p ≠ q):  log_p q + 3 log_q p = 4

                                          pq = 81                        (5)

(c)    3 log₂ x + 4 log₃ y = 10

   log₂ x – log₃ y = 1                                                (6)

10.  Solve:         (a) log_q 5 + 6 log₅ q = 5                                                (4)

                  (b) log₃ (5x + 12) + log₃ x = 2                                      (5)

11.  Given that log₉ 10 = k log₃ 10,

(a)   find the value of k.                                                                     (2)

      (b) Factorise completely

                                    4x log₃ x – 3 log₃ x + 16x log₉ 10 – 12 log₉ 10  (2)

(c)   Hence, solve the equation                                                          (5)

                        4x log₃ x – 3 log₃ x + 16x log₉ 10 – 12 log₉ 10 = 0       

                       

12.    Solve

(a)    log_p 343 = 3                                                                (2)

(b)   log₆(11q – 4) = 3                                                                     (2)

(c)    9 log_r 3 = log₃ r                                                                        (3)

(d)   Show that                                                                                (3)

           log₅ 3 + log₅ 6 + log₅ 9 + log₅ 12 + log₅ +15 = 6 log₅ 3 + 3 log₅ 2 + 1

(e)    Solve the equation                                                                   (3)

           log₅ 3 + log₅ 6 + log₅ 9 + log₅ 12 + log₅ +15 = 1 + log₅ x + log₅ x²

13.    (a) Solve the equations

(i)                  log_x 343 = 3                                                              (2)

(ii)                log₉(4y – 3) = 2                                                         (2)

(b)   Solve, to 3 significant figures, log_q 5 + 6 log₅ q = 5        (5)

(c)    Show that x log₂ x⁵ – log₂ x² ≡ (5x – 2) log₂ x.               (2)

(d)   Hence solve the equation x log₂ x⁵ – log₂ x² = 20x – 8   (4)

                                                           

14.    (a) Solve the equations log₄ 2 = p                                       (1)

Given that log₂ 3 = k log₄ 3

(b)   find the value of k                                                                   (2)

(d)      Show that                                                                               (4)

     5x log₄ x – 2 log₄ x – 10x log₂ 3 + 4 log₂ 3 = log₄ (x^{5x – 2}/3^{20x – 8})

(e)      Hence solve the equation                                                        (4)

5x log₄ x – 2 log₄ x – 10x log₂ 3 + 4 log₂ 3 = 0

15.    Solve

(a)    log_q 343 = 3                                                    (2)

(b)   log₄(5n + 9) = 3                                               (3)

(c)    log_m 4 + 8 log₄ m = 6                                       (6)

(d)   2 log₃ x - 3x log₃ x + 6x = 4                             (5)

16.  Solve

(a)      log_p 243 = 5                                                               (2)

(b)      log₄ (3q + 4) = 3                                                        (2)

f(x) = 2x log_x 3 – 5 log_x 9 – x + 5

(c)      Find the value of a and the value of b so that

         

          f(x) = (x – 5)(a log_x 3 – b)                                (3)

(d)      Hence solve the equation f(x) = 0                                (3)

17.    (a) Solve the equations

(i)                  log₅ 625 = x                                                              (2)

(ii)                log₃(5y + 3) = 5                                                         (2)

(b)   (i)       Factorise          5x ln x + 3 ln x – 10x – 6                     

(ii)      Hence find the exact solution of the equation

5x ln x + 3 ln x – 10x – 6 = 0                            (5)

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