Logarithms 21
Problem 1. Evaluate log 3 9.
Let x = log 3 9 then 3
x
= 9
from the definition
of a logarithm,
i.e.
3
x
= 3
2
from which, x = 2
Hence,
log 3 9 = 2
Problem 2. Evaluate log 10 10.
Let x = log 10 10 then 10
x
= 10
from the
definition of a logarithm,
i.e.
10
x
= 10
1
from which, x = 1
Hence,
log 10 10 = 1
(which may be checked
by a calculator)
Problem 3. Evaluate log 16 8.
Let x = log 16 8 then 16
x
= 8
from the definition
of a logarithm,
i.e. (2
4
)
x
= 2
3 i.e. 2
4x
= 2
3 from the laws of indices,
from which,
4x = 3 and x =
3
4
Hence,
log 16 8 =
3
4
Problem 4. Evaluate lg 0.001.
Let x = lg 0.001 = log 10 0.001
then 10
x
= 0.001
i.e.
10
x
= 10
−3
from which, x = −3
Hence,
lg 0.001 = −3 (which may be checked
by a calculator)
Problem 5. Evaluate ln e.
Let x = ln e = log e e then e
x
= e
i.e.
e
x
= e
1
from which, x = 1
Hence,
ln e = 1 (which may be checked
by a calculator)
Problem 6. Evaluate log 3
1
81
.
Let x = log 3
1
81
then 3
x
=
1
81
=
1
3 4 = 3
−4
from which, x = −4
Hence,
log 3
1
81
= −4
Problem 7. Solve the equation: lg x = 3.
If lg x = 3 then log 10 x = 3
and
x = 10
3
i.e. x = 1000
Problem 8. Solve the equation: log 2 x = 5.
If log 2 x = 5 then x = 2 5 = 32
Problem 9. Solve the equation: log 5 x = −2.
If log 5 x = −2 then x = 5 −2 =
1
5 2 =
1
25
Now try the following exercise
Exercise 11 Further problems on laws of
logarithms
In Problems 1 to 11, evaluate the given
expressions:
1. log 10 10000 [4]
2. log 2 16
[4]
3. log 5 125
[3]
4. log 2
1
8
[−3]
5. log 8 2
1
3
6. log 7 343 [3]
7. lg 100
[2]
8. lg 0.01 [−2]
9. log 4 8
1
1
2
10. log 27 3
1
3
11. ln e 2
[2]