FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 2.27- If p = ax, q = ay and a^4 = (p4y. q4x)x prove that xyz = 21.
- Prove that log3 3 log 8 log 125 3log6 log5+ − − = 2
COMMON LOGARITHM :
Logarithm to the base 10 is called common logarithm. For numerical calculations, common logarithm is
usually used. This system was first introduced by Henry Briggs.
In future, the base of the common logarithm will not be written. Thus log 4 will really log 10 4.
100 = 1 ∴ log 1 = 0
101 = 10 ∴ log 10 = 1
102 = 100 ∴ log 100 = 2 and so on.
Again since,
10 –1 = 101 = 0.1 ∴ log 0.1 = – 1
10 –2 = 1012 = 0.01 ∴ log 0.01 = – 2 and so on
2.4.1 ANTILOGARITHM
If 10x = N, i.e., if log N = x, then N is called the antilogarithm or antilog of x.
e.g : Since log 100 = 2 ∴ antilog 2 = 100
log 1000 = 3 ∴ antilog 3 = 1000
log 0.1 = 1 ∴ antilog 1 = 0.1
log 0.01 = 2 ∴ antilog 2 = 0.01 and so on.
Characteristics and Mantissa :
We know, log 100 = 2 and log 1000 = 3.
Now 517 lies between 100 and 1000.
i.e. 100 < 517 < 1000
or log 100 < log 517 < log 1000
or 2 < log 517 < 3, hence log 517 lies between 2 and 3.
In other words, log 517 = 2 + a positive proper fraction.
Again 0.001 < 0.005 < 0.01
or – 3 < log 0.005 < – 2, for log 0.001 = – 3, log 0.01 = – 2.
Hence log 0.005 is greater than – 3 and less than – 2 and is negative.
In other words, log 0.005 = – 3 + a positive proper fraction. Thus we see that logarithm of any number
consists of two parts, an integral part (positive or negative) and a fractional part.
The integral part is called characteristics and the fractional part is mantissa.