Logarithms

The idea of logarithms is that they are the reverse operation of exponents, that is the purpose of them. That is that if a^b = x, then, log_ax = b. John Napier was the first person to introduce logarithms in the 1600s and they rapidly came into use, they were especially useful for one fact about logarithms.

That is that log_a(xy) = log_a(x) +  log_a(y) . This made large multiplications a lot simpler and simply turned it into a problem of addition. And in fact we can prove this fact relatively simply, and I will do after proving one other property. That is that  log_a(x^c) = c log_a(x).

Let log_a(x) = b, if we multiply both sides by c and write log_a(x) = b in exponent form we get:
c log_a(x) = bc and a^b = x. If we raise both sides of a^b = x to the power of c we get:
a^bc =  x^c turning this into a log gives:
 log_a(x^c) = bc, and we know that bc = c log_a(x), so this means that:
 log_a(x^c) = c log_a(x), proving what we needed to know.

Now to prove that log_a(xy) = log_a(x) +  log_a(y):
Let m = log_a(x) and n = log_a(y) and then write these in exponent form 

x = a^m and y = aⁿ then we multiply these together to give:

x × y = a^m × aⁿ = a^m+n, we can now take log_a of both sides and evaluate 

log_a(xy) = log_a(a)^m+n, then we use what we proved previously to get:

log_a(xy)  = (m + n) log_aa, note that log_aa = 1 for all a.
log_a(xy)  = m + n, we know from our first line that  m = log_a x and n = log_a y so:
log_a(xy) = log_a(x) + log_a(y), proving what we want!

We can utilise the last two properties to also prove another property (we can also use the last method to prove it too, but this is far easier).

Consider log_a(x/y) and notice that this can be wrote as:

log_a(x × y^-1), which from using our last fact means that:
log_a(x × y^-1) = log_a(x) + log_a(y^-1), using the first property (log_a(x^c) = c log_a(x)) this means that:
log_a(x/y) = log_a(x) - log_a(y), providing another identity.

And that really is it for the basics of logarithms, you can do a lot with them so make sure you can understand these simply properties, they will make other problems a lot, lot simpler.
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