Why there are two kinds of Logarithm?
Where to use them separately ?
common logarithm vs natural logarithm?
When solving a problem such as 5x = 25, will I get the same answer when I use common logarithm as when I use natural logarithm? and how that come?
Any logarithm will do because 5=b^(log_b 5) where b>0 and b not 1. Thus,
5^x = 25
becomes
x log_b 5 = log_b 25 --> x = log_b 25 / log_b 5
No matter what b you use this quotient has the same value. It is a matter of convenience. When solving 5^x =25, log_5 gives you
x log_5 = log_b 25
or
x(1)=log_5(5^2)=2--> x=2.
logarithms are just inverses of exponentials. The b just identifies which logarithm f(x)=log_b x and which exponential f^(-1)(x)=b^x
There are not just two kinds of logarithms. There are infinitely many bases you could use. Depending on your study, you may be commonly using different bases. The point of logarithms are to reduce multiplication and division operations down to addition and subtraction operations; which are far easier to calculate for large values. They also capture tendencies quite well like asymptotics.
A property that makes logarithms handy are that they are one-to-one. There will be a unique y, so that b^y=x. We write this as log_b(x)=y; that's the definition of a logarithm. What you call "different" logarithms, they are just different bases. Your best friend if you are interested in different bases and how they relate is the change of base property of a logarithm; which it is straightforward to find on the Internet (and has been given above by other answers).
In terms of asymptotics (as seen and used in number theory or analysis of algorithms), the one powerful thing is that bases of logarithms when converted only differ by a constant amount. So it is very common to not even care about the base in some disciplines unless you want to be very precise.
I agree with those that have said that they are indefinitely number of different types of logarithms, but not all of them have perhaps practical application; for this reason perhaps looks that we have only a few types of different logarithms.
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