1. Common Logarithms

2. In fact, any positive number can be expressed as a power of 10.
10  101
100  102
1000  103
 104
5 ≈ 10?
12 ≈ 10?
In fact, any positive number can be expressed as a power of 10.

3. 10  101
100  102
1000  103
 104
5 ≈ 10?
12 ≈ 10?
5 ≈ 10
0.699
12 ≈ 10
1.079

4. When x is expressed as the base 10,
x = 10y
the index y is called the logarithm of x to the base 10.
It is also called the common logarithm of x,
denoted by log10 x.
Symbolically:
The base 10 may be omitted.
y = log10 x
So, we have:
If x = 10y, then y = log x.

5. x = 10y > 0 for all values of y
For example: ∵
100  102 ∴
Common logarithm of 100
= 2
log 100 = 2 ∵
5 ≈ ∴
Common logarithm of 5
≈ 0.699
log 5 ≈ 0.699
Note:
log x is undefined for x  0.
x = 10y > 0 for all values of y
e.g. log 0 and log (3) are undefined.

6. Exponential form (x = 10y) Logarithmic form (log x = y)
= 101
◄ log 101 = 1
log 1 =
1 = 100
◄ log 100 = 0
0.1
log 0.1 = 1
= 101
◄ log 10–1 = –1
From the above table, we notice that:
log 10 = 1 and log 1 = 0

7. In fact, the following is also true:
If y = log x, then x = 10y.
For example:
log x = 2
x = 102
= 100
log x = 0.5
x = 100.5
≈ 3.16

8. Follow-up question
Find the values of the following common logarithms without using a calculator.
(a) log 1000
∵ = 103
∴ log 1000 = 3
◄ If x = 10y, then log x = y.
(b) log 0.01
∵ = 102
∴ log 0.01 = 2

9. We can also find the values of common logarithms of positive numbers with a calculator.
By keying in:
gives
∴ log 5 = (cor. to 3 sig. fig.)

10. Properties of Common Logarithms
For any M, N > 0, we have:
1. log (MN) = log M + log N
2. log = log M  log N
3. log Mn = n log M
◄ n is a real number.
The proof for these properties can be viewed in Proof of Common Logarithms.ppt.

11. without using a calculator.
Find the value of
log 2 + log 50
without using a calculator.
log 2 + log 50
= log (2  50)
log M + log N = log (MN)
= log 100
= log 102
= 2
Definition of common logarithms

12. Let me try to find the value of log 40  log 4
without using a calculator.
log 40  log 4
log M  log N = log =
= log 10
= 1

13. Follow-up question Find the value of without using a calculator.  8 3
log 81
Find the value of without using a calculator. 3
log 81 2 1 4 3
log =
◄ Express 81 and as powers of 3. 3 3
log 2 1 4 =
◄ log Mn = n log M
 8