Logarithms

Why logarithms:

We know that 32=25$32=2^5$.  ie., 32 can be represented as a power of 2.  But how do we represent 32 as some power of 10?
This is where logarithms comes.  32 when represented a power of 10 is equal to log1032=1.5051${\log }_{10}32=1.5051$.  i.e., 32=101.5051$32={10}^{1.5051}$

Properties of Logarithms:

1. loga1=0${\log }_{a}1=0$ because 0 is the power to which a must be raised to obtain 1.
2. logaa=1${\log }_{a}a=1$ because 1 is the power to which a must be raised to obtain a.
3. logaaN=N${\log }_a{a}^N=N$ because N is the power to which a must be raised to obtain aN$a^N$
4. alogab=b$a^{{\log }_{a}b}=b$ because logab${\log }_{a}b$ is the power to which a must be raised to obtain b.
5. alogbc=clogba$a^{{\log }_{b}c}=c^{{\log }_{b}a}$
6. logabM=1blogaM${\log }_{a^b}M={\displaystyle \frac{1}{b}{\log }_{a}M}$
7. logbM=logaMlogab${\log }_{b}M={\displaystyle \frac{{\log }_{a}M}{{\log }_{a}b}}$
8. logab=1logba${\log }_{a}b={\displaystyle \frac{1}{{\log }_{b}a}}$
9. logab.logbc=logac${\log }_{a}b.{\log }_{b}c={\log }_{a}c$

Laws of logarithm:
1. Product rule: logaMN=logaM+logaN${\log }_{a}MN={\log }_{a}M+{\log }_{a}N$
2. Division rule = logaMN=logaM−logaN${\log }_a{\displaystyle \frac{M}{N}={\log }_{a}M-{\log }_{a}N}$

Remember: 
loga(b+c)≠logab+logac${\log }_a(b+c)\ne {\log }_{a}b+{\log }_{a}c$

Note: When no base is given we generally assume the base as 10.  These are called common logarithms
When "e" is a base then we call them as natural logarithms.

Characteristic of logarithm:
The characteristic of the logarithm of a number greater than one is one less than the number of digits in it.
Eg: characteristic of 98765 = 4 so total digits in the given number is 5.
We know that log 100 = log102=2log10=2.00$\log {10}^2=2\log 10=2.00$.  As the characteristic of 100 is 2, then total digits in 100 are 3.
log 10 = log101=1log10=1$\log {10}^1=1\log 10=1$ So total digits are 2 in 10.
log 99 = 1.9956 so total digits are 1 + 1 = 2.

Working Rule to find the number digits in ab$a^b$ format number: 
1. Calculate the logarithm (you will get some positive number)
2. Adding one to the intezer part will give you the number of digits in that original number