Logarithm form
In this definition y = log_{b } x is called the logarithm form
Exponential form
In this definition b ^{y}= x is called the exponential form.
Note that the requirement that x >0 is actually a result of the fact which we are also requiring b = 0 . If you think regarding it, it will make sense. We are raising positive number to an exponent & thus there is no way that the result can probably be anything other than another positive number. It is extremely important to remember that we can't take the logarithm of zero or a negative number.
Now, let's address the notation utilized here as i.e. the biggest hurdle usually that students have to overcome before beginning to understand logarithms. Firstly, the "log" part of the function is just three letters which are utilized to indicate the fact which we are dealing with a logarithm. They are not variables & they aren't signifying multiplication. They are only there to tell us we are dealing along with a logarithm.
Next, the b i.e. subscripted on the "log" part is there to tell us what the base is as it is an significant piece of information. Also, in spite of what it might look like there is no exponentiation in the logarithm form above. It may look like we've got b^{x} in that form, although it isn't. It just looks like which might be what's happening.
It is significant to keep the notation with logarithms straight, if you don't you will determine it very hard to understand them and to work with them.