Twisting the brain in maths? Here you see how maths is interesting in this aspect!
The study of Logarithm is fun and exciting. But let's start off with explanation of the base.
Logarithm term is represented as logaN.
The small "a" is called the "base". It is not necessary that all term of logarithm uses the same base. It can be log2X, log10X , etc.
What if we want a certain base but is not given in the logarithmic problem?
Let's explain with an example. Simplify log3X + log9Y.
Here we want the base to be the same so that we can combine the 2 logarithmic terms using the Product Law. How?
Make use of the "Change of Base" method ==> logaN = (logbN ) / logba
We can change log9Y (by choice only) to base 3 to be same as first logarithmic term.
Using the Change of Base technique, log9Y becomes ( log3Y) / log39.
Thus log3X + log9Y = log3X + (log3Y )/ log39 = log3X + (log3Y) / 2
NOTE: log39 = log 3 32 = 2 log33 = 2(1) using Power Law of Logarithm.
log3X + log9Y = log3X + log3Y(1/2) = log3(XY1/2) using Product Law.
From the above example, we can see the advantage of the Logarithm "Change of Base" technique to simplify a logarithmic expression. Without applying the method, there is no way we can combine them as they are of different bases!
New challenge: Simplify logx7 / logx3 +2 log27 / log23
Answer: 3 log 37. You got it?