Logarithm - Change of Base Technique

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 log_aN.

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 ==> log_aN = (log_bN ) / log_ba

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(XY^1/2) 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 log_x7 / log_x3 +2 log₂7 / log₂3

Answer: 3 log ₃7. You got it?

:-)