Question 29231
First of all you should learn the correct definition of logarithm  of a number.
The logarithm of a real number N is defined only for N strictly positive.
(this means the logarithm of N is not defined for N=0 and not defined for N<0)
Given a positive number N the logarithm of N to a given base b is defined 
as the power p to which the base has to be raised to give the number
log N to base b is p implies and is implied by  N=(b)^p  ----(*)
examples: 
1)log(100) to the base  10
Since  100=(10)^2   
we write log(100) to base 10 is 2
[here N=100, b=10, so you ask the  question 10 raised to what is 100 and you get (10)^2 =100 
and so the power is 2 ] 
2)log(81) to base 3
Since (3)^4 = 81, we write log(81) to the base 3 is 4 
[here N=81, b=3, so you ask the  question 3 raised to what is 81 and you get (3)^4 =81 
and so the power is 4 ] 
Got it!
3)If you are given (2)^5=32, how do you express  it in the logarithmic language?
Notice that  our N=32, b=2 and p=5  [look at  (*)!(asking you to see stars!)]
Therefore  log(32) to the base 2 is 5
4)If you are given (5)^3= 125, how do you express  it in the logarithmic language?
Notice that  our N=125, b=5 and p=3  [look at  (*)]
Therefore  log(125) to the base 5 is 3
If it is confusing, keep chanting  the definition (*)
Cool is n't it?

Four important initial formulae
caution:Every formula in logarithm  for any consideration is  applicable  only for a  particular given base.[If you are changing base  then you have to apply the formula for the change of base]
i)for a given base b
   log(m) +log(n) = log(mn)----(1)
   log(m) -log(n) = log(m/n)----(2)
   n[log(m)]   =  log[(m)^n]----(3)
  log(m) to a base (b) given by {[log(m) to a base (c)]/[log(b) to the base (c)]
  where c is the new base.
Note: logarithm of any number (m) to the same base (m) is 1
as(m)^1=m
Note: NO FORMULA for log(m)Xlog(n).
That is for a given base b there is no formula for multiplication between log(m) and log(n) 
Note: log(m) to the base (a) multiplied by log(a) to the base (n)
that loga(m) X logn(a) = logn(m)  -----(*)
and this should not be mistaken for a stadard formula for multiplication between two logartihms. What you must NOTE here is though two logarithms are multiplied they are not with respect to the same base and all formulae in logarithms are dealing in 
logarithm of a product = sum of logarithms, (for a given base b)
logarithm of  a quotient = difference between the logarithm  
(for a given base b) and the other two formulae too are for a given base b
and in the star (*) above one logarithm has for its base a and the other has for its base n
This (*) if you look closely,is the result of change of base formula
Actually this is the result of logn(m)= loga(m)/loga(n) 
(Change of base rule) (and here the new base is a)
Multiplying by loga(n)
we get  (*)

Examples of formula (1)
When it is mentioned "common logarithms"  the base  is 10
You use this  for numerical problems
When it is mentioned "natural logarithms"  the base  is e (the irrational number more than  2 and less than 3 called  the Napierian base named after the  mathematician Napier)
You use  this for problems in Calculus.