1

Miscellaneous Exercise 1
1. (a) Rationalizing each fraction as a separate
item
5 5
(i)
5 3 5 3
x
(ii) 1
x
1
x
(b) Express the following surd number with
rational denominators
(i)
(ii)
6
6
3
3
1
2
2
2 2
1
x
x
1
log 3 x
2. (a) Evaluate (i)
log
x
8
(ii) log 3
2log 15 log 0.14 10 log 7
2 6 7 7 98
3. Show that log b
a
1
log
Using the result to prove that
log b
log c log c
a
b
4. Express the following as a single logarithm
1
3
(i) 2log a log 10b 3
log 2 5 4
2
(ii) log 6 8log 16 9log 18
7 3 2 10
5.(a) Simplify the expression
(i) 5
4 20 8
(ii)
3
9
5n
a
b
3n1
2n
2n2
6
2n3
n2
6
2 6. (a) If a log x pq,
b log p rq,
c log r pq
prove that a b c abc 2
(b) Show that log blog
a 1and hence
a
5 64
evaluate log 2log 4 24
7. Solve simultaneous equations
x2
y1
(a) 5 7 3468 and 7 5 76
(b) 6log
3 x 6log 27 y 7 and
log x 4log y 9 .
4 9 3
8. Solve (a)
2
a
b
y x
log 9 x log 9 x log 9 x log 9 x 5
3
4
3
(b) log 4 x log 4 7 .
2
9.a) Express
5
(i) log a x log ax
1 log a x
2
and(ii) 7log
a 2 3log a 12 5log a 3 as a
single logarithm.
1
(b) Prove that log 2 10 and
log 2
evaluate log e
100 correct to 3 significant
figures taking e as 2.718.
log 10 a
10.(a) Prove that log e a ,hence show
log e
that log 718 2 ln 2
2.
(b) (i) Prove that
10
10
a
log log c a log c b
b
2
(ii) Find x if log 8 log 2 1
11. (a) Solve the simultaneous equations
p and 3 2 2 2 16
2 4 12
(b) If
q
Find x
x
p
x
2q
log x log 9 x log 81 x
3
1
3 5
(c) Simplify
2 5
16
28
12. (a) (i) Find log 8
64 4 without using
calculator or tables.
2 3
(ii) Simplify log
q log
p
p
(b) Find the values of t and m such that
(i) log 9 t log 9 m 1and
log 9 t log 9 m log 91.5
log
8
t 8 log
8
t 8
(ii)
3
13. (a) Given that
log
log
10
10
7
3
9
5
q
, solve for
p and q the simultaneous equations
p 3 1
3
q
p 1
7 and 9
(b)(i) Prove that log
x
2q1
63
q log
r
.
p log
r
pq