112 Basic Engineering Mathematics
Here are some worked problems to help understanding of logarithms.
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
using 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
using 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 Practice Exercise
Practice Exercise 59 Laws of logarithms
(answers on page 346)
In problems 1 to 11, evaluate the given expressions.
1. log 10 10000
2. log 2 16
3. log 5 125
4. log 2
1
8
5. log 8 2
6. log 7 343
7. lg 100
8. lg 0.01
9. log 4 8
10. log 27 3
11. ln e 2
In problems 12 to 18, solve the equations.
12. log 10 x = 4
13. lg x = 5
14. log 3 x = 2
15. log 4 x = −2
1
2
16. lg x = −2
17. log 8 x = −
4
3
18. ln x = 3