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Form 2 Questions and Answers on Indices and Logarithms
Simplify the following:
(a)
r
3
×
r
2
t
2
×
r
-
3
t
-
2
(b)
a
n
×
a
m
a
-
n
×
a
-
m
(c)
x
2
y
3
×
x
4
y
2
x
3
y
4
(d)
a
4
b
3
×
a
2
c
4
×
b
-
2
c
3
a
-
3
b
3
c
3
(5m 47s)
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1.
Solve for x in:
9
x
+
3
2
x
-
1
=
53
2.
Solve the equation :
9
x
+
1
+
3
2
(
x
+
1
)
=
36
3.
Find the value of x in the following equation:
49
x
+
1
+
7
2
x
=
350
4.
Given that
9
2
y
×
2
x
=
72
, Find the value of x and y
5.
Two numbers p and q are such that
p
3
×
q
=
189
. Find p and q
6.
Given that
9
4
3
=
3
n
,find the value of n
7.
Find the value of x which satisfies the equation:
16
x
2
=
8
4
x
-
3
8.
Find the value of m in the following equation:
(
1
27
)
m
×
(
81
)
-
1
=
243
9.
Solve for x in the equation:
(
81
2
x
×
27
x
)
÷
9
x
=
729
10.
Without using a calculator or mathematical tables, evaluate
(
27
)
2
3
×
(
81
16
)
-
1
4
11.
Simplify
(
27
2
3
÷
2
4
)
÷
32
-
3
5
12.
Without using logarithm tables or calculators, evaluate,
(
64
-
1
2
×
27000
2
3
)
÷
(
2
-
4
×
3
0
×
5
2
)
13.
Solve for x:
4
x
+
1
=
32
14.
Solve for x in the equation:
32
(
x
-
3
)
×
8
(
x
+
4
)
=
64
÷
2
x
15.
Simplify:
(
2
x
×
5
2
x
÷
2
-
x
)
1
2
16.
Find the value of y in the equation:
243
×
3
2
y
729
×
3
y
÷
3
(
2
y
-
1
)
=
81
17.
Simplify:
243
-
2
5
×
125
2
3
9
-
3
2
18.
Simplify:
25
3
4
×
0.9
2
×
2
2
5
5
2
×
3
3
19.
Given that
6
2
(
n
-
3
)
=
7776
, find the value of n.
20.
Given that P=3y express the equation
3
2
y
-
1
+
2
×
3
y
-
1
=
1
in terms of P. Hence or otherwise find the value of y in the equation
3
2
y
-
1
+
2
×
3
y
-
1
=
1
21.
Use logarithms to evaluate:
3
√
(
35.6
×
0.0613
2
)
22.
Evaluate:
1.34
(
5.24
0.8
)
×
0.0029
23.
Use mathematical tables to evaluate:
√
(
2.935
×
0.0765
32.74
)
24.
Use logarithms to evaluate:
3
√
(
7.08
76.8
×
7.034
)
25.
Use mathematical tables to evaluate:
√
0.0645
0.0082
26.
Use logarithms to evaluate
4
√
4.562
×
0.038
0.82
27.
Use logarithms to evaluate
(
0.07284
)
2
3
√
0.06195
28.
Use logarithms to evaluate
3
√
36.15
×
0.02573
1.938
29.
Use logarithms to evaluate:
(
1934
)
2
×
√
0.00324
436
30.
Use logarithms to evaluate:
55.9
÷
(
0.2621
×
0.01177
)
1
5
31.
Use logarithms to evaluate
(
6.79
×
0.3911
log
5
)
3
4
32.
Use logarithms to evaluate
3
√
1.23
×
0.0089
76.54
33.
Use logarithms to evaluate:
(
3.256
×
0.0536
)
1
3
34.
Use logarithms to evaluate
(
0.0056
)
1
2
1.38
×
27.42
35.
Use logarithm tables to evaluate
2347
×
0.04666
3
√
0.0924
36.
Use logarithms to evaluate
34.33
√
5.25
×
0.042
37.
In this question, show all the steps in your calculations, giving your answer at each stage. Use logarithms, correct to 4 decimal places, to evaluate:
3
√
36.72
×
(
0.46
)
2
185.4
38.
Use logarithm tables to evaluate:
(
0.032
×
14.26
0.006
)
2
3
39.
Use logarithms, correct to 4 decimal places, to evaluate:
3
√
83.46
×
0.0054
1.56
2
40.
Use logarithms, correct to 4 decimal places, to evaluate
3
√
1.794
×
0.038
1.243
41.
Find the numerical value of: (a)
9
1
2
(b)
25
-
1
2
(c)
36
3
2
(d)
8
-
5
3
42.
Find the numerical value of each of the following: (a)
(
3
-
2
)
3
(b)
(
49
2
)
-
1
2
(c)
(
10
3
)
5
3
43.
Simplify each of the following expressions leaving your answer in index form. (a)
3
3
×
3
4
(b)
5
2
×
5
-
3
(c)
3
3
2
×
3
-
3
4
×
3
1
4
(d)
6
2
×
7
-
4
×
(
8
-
2
)
2
×
6
3
×
7
2
×
8
4
44.
Simplify the following expressions leaving your answer in index form. (a)
512
2
3
×
8
1
3
÷
2
6
(b)
2
-
2
×
(
2
2
)
-
6
2
-
4
×
2
-
6
(c)
(
6
7
)
-
2
×
(
7
6
)
-
2
45.
Simplify the following: (a)
r
3
×
r
2
t
2
×
r
-
3
t
-
2
(b)
a
n
×
a
m
a
-
n
×
a
-
m
(c)
x
2
y
3
×
x
4
y
2
x
3
y
4
(d)
a
4
b
3
×
a
2
c
4
×
b
-
2
c
3
a
-
3
b
3
c
3
46.
Simplify: (a)
25
y
9
k
-
7
81
y
7
k
9
(b)
3
-
4
a
4
b
6
3
a
5
b
8
(c)
(
a
-
n
)
m
×
a
-
2
m
a
m
n
×
a
2
m
47.
Simplify the expressions: (a)
(
9
a
2
)
1
3
6
4
×
(
4
b
2
)
1
3
5
a
(b)
3
√
27
x
3
y
9
x
6
y
3
(c)
5
√
32
y
10
k
15
2
10
4
√
(
16
y
12
k
16
2
4
)
48.
Solve each of the following equations: (a)
(
3
2
x
)
3
=
3
4
×
3
8
(b)
(
7
5
)
x
=
(
7
4
)
x
÷
7
2
(c)
64
2
y
÷
16
y
128
y
×
4
2
y
49.
Find the logarithms to base of 10 of: (a) 0.149 (b) 0.0000784 (c) 8.2 (d) 475000
50.
Find the antilogarithms of the following numbers. (a)
ˉ
1
. 1080 (b)
ˉ
4
. 8938
51.
Calculate: (a)
ˉ
2
. 34 +
ˉ
4
. 30 (b)
ˉ
3
. 79 -
ˉ
2
. 24
52.
Use logarithms and antilogarithm tables to evaluate: (a) 251 × 367 (b) 7.32 × 199
53.
Use logarithms and antilogarithm tables to solve: (a) 1500 ÷ 750 (b) 43.1 ÷ 3.17
54.
Use logarithms and antilogarithm tables to evaluate the following: (a)
291
×
681
372
(b)
1024
×
6561
4096
×
729
55.
Use logarithm table to find (i) the square root and (ii) the cube root of: (a) 478 (b) 0.0067
56.
Use tables to evaluate: (a)
√
3.142
×
2.718
6.49
×
81.2
(b)
41.56
×
52.3
√
42.88
57.
Find the numerical value of: (a)
36
-
1
2
(b)
35769
0
(c)
81
1
2
(d)
27
-
1
3
(e)
16
5
2
(f)
100
9
2
(g)
64
4
3
(h)
256
3
4
(i)
729
1
3
(j)
(
216
2
)
1
3
(k)
243
2
3
(l)
4096
-
3
4
(m)
64
-
1
6
58.
Simplify each of the following expressions. You may leave your answer in index form: (a)
2
3
×
2
5
×
2
-
4
(b)
20
-
3
×
25
2
×
20
3
×
25
-
4
(c)
7
-
3
×
8
4
×
7
2
×
8
-
3
(d)
27
1
3
×
9
1
2
(e)
(
3
4
)
2
×
(
3
4
)
3
(f)
20
-
2
×
20
-
2
×
20
-
2
20
(g)
2
-
6
×
5
1
2
×
2
7
÷
2
-
4
59.
Simplify:
(
a
)
x
2
×
x
4
(
b
)
y
3
×
y
6
(
c
)
n
13
×
n
-
7
(
d
)
a
9
×
a
3
(
e
)
a
-
2
×
3
a
4
(
f
)
(
2
n
)
3
×
(
2
n
)
5
(
g
)
x
16
÷
n
12
(
h
)
a
15
÷
a
14
(
i
)
a
2
×
b
2
×
a
4
60.
Simplify: (a)
b
2
×
x
2
×
b
11
(b)
b
3
×
n
2
×
b
0
×
n
3
(c)
a
5
×
c
2
×
a
3
×
b
3
×
a
4
(d)
a
-
2
×
b
2
×
a
-
8
×
b
-
4
(e)
x
2
y
2
×
x
2
y
5
×
x
-
3
y
-
9
(f)
(
x
y
t
)
n
÷
(
x
y
t
)
2
(g)
a
x
×
a
2
x
a
x
×
a
x
×
a
x
61.
Simplify: (a)
12
a
5
b
3
4
a
3
b
(b)
a
-
a
×
b
-
a
×
c
-
a
a
-
2
a
×
b
-
2
a
×
c
-
2
a
(c)
r
s
2
t
3
×
r
4
s
5
t
-
6
×
r
7
s
8
t
-
9
r
10
s
11
t
-
12
×
r
13
s
-
14
t
15
(d)
(
16
a
2
)
1
2
×
(
36
a
4
)
-
1
2
2
a
1
2
×
5
a
3
2
×
8
a
9
4
62.
Solve each of the following equations: (a)
(
3
2
x
)
4
=
81
(b)
4
5
x
÷
(
2
3
x
)
2
=
256
(c)
9
2
x
=
729
(d)
2
8
x
=
512
(e)
(
7
4
)
2
x
=
(
7
4
)
3
(f)
(
5
3
)
a
×
(
5
8
)
a
=
5
72
63.
Solve each of the following equations: (a)
9
4
x
÷
3
2
x
=
2187
(b)
(
5
2
c
)
3
=
(
5
4
)
c
×
625
(c)
2
x
4
=
8
(d)
3
(
2
x
-
5
)
=
27
(e)
3
4
x
÷
3
-
7
=
3
15
(f)
2
×
3
x
=
162
64.
Write in form of
a
n
where a=2 (i) 2 (ii) 4 (iii) 1 (iv)
1
2
(v)
1
4
65.
Write in form of
a
n
where a=3 (i) 27 (ii) 9 (iii) 3 (iv) 1 (v)
1
3
(vi)
1
9
(vii)
1
27
66.
Write in form of
a
n
where a=10 (i) 1000 (ii) 100 (iii) 10 (iv) 1 (v)
1
10
(vi)
1
100
(vii)
1
1000
67.
Copy and complete the table below:
68.
Write in logarithmic form: (a)
3
2
=
9
(b)
2
4
=
16
(c)
3
3
=
27
(d)
2
5
=
32
(e)
3
4
=
81
(f)
5
3
=
125
(g)
10
0
=
1
(h)
2
10
=
1024
(i)
a
n
=
b
69.
Write each of the following in index form: (a)
log
2
8
=
3
(b)
log
4
16
=
2
(c)
log
5
125
=
3
(d)
log
10
8
=
x
(e)
log
b
a
=
c
(f)
log
3
27
=
3
(g)
log
6
216
=
3
(h)
log
x
40
=
y
(i)
log
y
x
=
2
(j)
log
10
10000
=
4
(k)
log
4
6
=
y
(l)
log
2
16
=
4
70.
Use logarithm table to express each of the following numbers in the form
10
x
. (a) 681.4 (b) 1.47 (c) 4.73 (d) 7.25 (e) 9.83 (f) 5.672 (g) 8.137 (h) 3.142 (i) 2.718
71.
Use logarithm table to express each of the following numbers in the form
10
x
. (a) 3.333 (b) 12.3 (c) 59.7 (d) 82.9 (e) 72 (f) 96.1 (g) 431.5 (h) 7924 (i) 1025
72.
Use logarithm table to express each of the following numbers in the form
10
x
. (a) 1913 (b) 4937 (c) 273.7 (d) 3,910,000 (e) 958,312
73.
Find the logarithms to base 10 of: (a) 0.2843 (b) 0.3520 (c) 0.4286 (d) 0.0694 (e) 0.0485 (f) 0.0239 (g) 0.000376 (h) 0.0000523
74.
Find the antilogarithms of the following numbers: (a)
ˉ
1
.2480
(b)
ˉ
2
.3927
(c)
ˉ
2
.5403
(d)
ˉ
3
.6503
75.
Find the antilogarithms of the following numbers: (a)
ˉ
3
.9750
(b)
ˉ
2
.5658
(c)
ˉ
5
.4533
(d)
ˉ
3
.6821
76.
Calculate: (a)
ˉ
2
.37
+
1.20
(b)
ˉ
5
.63
+
ˉ
2
.57
(c)
ˉ
7
.12
-
ˉ
5
.54
(d) 5.27 -
ˉ
2
.73
77.
Use logarithm and antilogarithm tables to evaluate: (a)
4192
×
3078
(b)
21.47
×
362.1
78.
Use logarithm and antilogarithm tables to evaluate: (a)
26.1
×
91.2
×
45.7
(b)
33.2
×
172
×
44.32
79.
Use logarithm and antilogarithm tables to evaluate: (a)
1527
×
3196
×
4157
(b)
7.312
×
49.45
×
157.2
80.
Use logarithm and antilogarithm tables to evaluate: (a)
16.31
×
152.1
×
3290
(b)
100
×
245
×
175
×
396
81.
Use logarithm and antilogarithm tables to evaluate: (a)
2145
÷
560
(b)
412
÷
241
(c)
882
÷
144
(d)
3612
÷
452
82.
Use logarithm and antilogarithm tables to evaluate: (a)
1111
÷
222
(b)
3.142
÷
2.718
(c)
9250
÷
4312
(d)
6380
÷
2137
83.
Use logarithm and antilogarithm tables to evaluate: (a)
634
×
436
×
688
784
(b)
3041
×
3211
2112
84.
Use logarithm and antilogarithm tables to evaluate: (a)
788
×
576
675
(b)
294
+
578
368
-
275
85.
Use logarithm and antilogarithm tables to evaluate: (a)
839
+
672
762
+
393
(b)
3267
-
37
2236
+
89
86.
Use logarithm and antilogarithm tables to evaluate: (a)
29
2
×
33
3
64
4
(b)
18
2
×
391
4
15
3
×
56
4
87.
Use logarithm and antilogarithm tables to evaluate: (a)
876.9
61.2
×
3.85
(b)
518.3
29.5
×
714
(c)
48.35
×
125.3
39.3
×
50.4
88.
Use logarithm and antilogarithm tables to evaluate: (a)
3.74
×
7.82
5.4
(b)
399.6
15.2
×
4.83
×
1.98
(c)
734.4
25.1
×
6.34
×
3.4
89.
Use logarithm and antilogarithm tables to evaluate: (a)
94.7
×
16.45
12.5
×
8.93
(b)
(
4.48
)
3
(
3.16
)
2
(c)
(
171.5
)
3
56.3
×
26.98
(d)
(
79.36
)
3
8.2
×
(
9.2
)
2
90.
Use logarithms to evaluate: (a)
0.9063
×
3.387
(b)
0.5060
×
0.05707
(c)
36.65
×
0.4163
×
0.007022
(d)
26.68
÷
255.4
91.
Use logarithms to evaluate: (a)
3.404
÷
628
(b)
0.0075
×
0.8181
0.00509
(c)
77.9
0.988
×
9100
92.
Use logarithms to evaluate: (a)
2.89
×
5.27
62.25
×
1.908
(b)
34.53
×
361.6
343.7
×
615.8
(c)
91.3
÷
18.26
75.4
÷
12.09
93.
Divide each of the following logarithms by 2. (a) 0.8938 (b) 1.8624 (c)
ˉ
2
.9754
(d)
ˉ
1
.7076
(e)
ˉ
1
.8538
(f)
ˉ
2
.3502
(g)
ˉ
2
.1644
(h)
ˉ
3
.4928
(i)
ˉ
3
.6946
(j)
ˉ
4
.7938
94.
Divide each of the following logarithms by 3. (a) 0.8938 (b) 1.8624 (c)
ˉ
2
.9754
(d)
ˉ
1
.7076
(e)
ˉ
1
.8538
(f)
ˉ
2
.3502
(g)
ˉ
2
.1644
(h)
ˉ
3
.4928
(i)
ˉ
3
.6946
(j)
ˉ
4
.7938
95.
Divide each of the following logarithms by 4. (a) 0.8938 (b) 1.8624 (c)
ˉ
2
.9754
(d)
ˉ
1
.7076
(e)
ˉ
1
.8538
(f)
ˉ
2
.3502
(g)
ˉ
2
.1644
(h)
ˉ
3
.4928
(i)
ˉ
3
.6946
(j)
ˉ
4
.7938
96.
Use logarithm table to find (i) the square root and (ii) the cube root of: (a) 2461 (b) 3572 (c) 4683 (d) 0.0346
97.
Use logarithm table to find (i) the square root and (ii) the cube root of: (a) 0.00457 (b) 0.00072 (c) 78.039 (d) 361.472
98.
Use tables to evaluate: (a)
2
√
(
3.45
×
16.7
31.5
)
(b)
3
√
(
1.794
×
0.038
12.43
)
(c)
3
√
(
39.51
×
725
0.758
)
99.
Use tables to evaluate: (a)
3
√
(
4862
×
725
6437
×
1024
)
(b)
4
√
(
6978
×
25.1
132.7
)
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