Everything Maths Grade 12

CHAPTER 2. LOGARITHMS 2.12

2. Show that
   loga

�√b

x

�

=

loga(x)

b

3. Without using a calculator show that:

log

75

16

− 2 log

5

9

+ log

32

243

= log 2

4. Given that 5 n= x and n = log 2 y
   (a) Write y in terms of n
   (b) Express log 84 y in terms of n
   (c) Express 50 n+1in terms of x and y


5. Simplify, without theuse of a calculator:

(a) 8

2

(^3) + log 232
(b) log 39 − log 5

√

5

(c)

�

5

4 −^1 − 9 −^1

�^1

2

+ log 392 ,^12

6. Simplify to a single number, without use of acalculator:

(a) log 5 125 +

log 32− log 8

log 8

(b) log 3− log 0, 3

7. Given: log 3 6 = a and log 6 5 = b
   (a) Express log 32 in terms of a.
   (b) Hence, or otherwise, find log 310 in terms of a and b.


8. Given: pqk= qp−^1

Prove: k = 1− 2 logqp

9. Evaluate without using a calculator: (log 7 49)^5 + log 5

�

1

125

�

− 13 log 91

10. If log 5 = 0, 7 , determine, without using a calculator:
    (a) log 25
    (b) 10 −^1 ,^4


11. Given: M = log 2 (x + 3) + log 2 (x− 3)
    (a) Determine the values of x for which M is defined.
    (b) Solve for x if M = 4.


12. Solve:

�

x^3

�log x

= 10x^2 (Answer(s) may be left insurd form, if necessary.)

13. Find the value of (log 27 3)^3 without the use of a calculator.


14. Simplify By using acalculator: log 4 8 + 2 log 3

√

27

15. Write log 4500 in terms of a and b if 2 = 10aand 9 = 10b.


16. Calculate:

52006 − 52004 + 24

52004 + 1

17. Solve the followingequation for x without the use of a calculator and using the fact
    that

√

10 ≈ 3 ,16 :

2 log(x + 1) =

6

log(x + 1)

− 1

18. Solve the followingequation for x: 66 x= 66 (Give answer correctto two decimal
    places.)