Principles of Mathematics

Logarithm - Use of Change of Base Method

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 method,   log₉Y    becomes   ( log₃Y) / log₃9. 

Therefore   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 usefulness of the Logarithm “Change of Base” method to simplify a logarithmic expression.  Without this, there is no way we can combine them as they are of different base!

Another challenge:    Simplify  log_x5 / log_x3   +   log₂5 / log₂3

Answer:   2log ₃5.

Article Submitted by : Lim Ee Hai

Article Source :    http://www.limeehai.com