The study of Logarithm is fun and exciting. But let's start off with explanation of the base.

Logarithm term is represented as log

_{a}N.

The small "a" is called the "base". It is not necessary that all term of logarithm uses the same base. It can be log

_{2}X, log

_{10}X , etc.

What if we want a certain base but is not given in the logarithmic problem?

Let's explain with an example. Simplify log

_{3}X + log

_{9}Y.

Here we want the base to be the

**so that we can combine the 2 logarithmic terms using the Product Law. How?**

*same*Make use of the "

**Change of Base**" method ==>

**log**

_{a}N = (log_{b}N ) / log_{b}aWe can change log

_{9}Y (by choice only) to base 3 to be same as first logarithmic term.

Using the Change of Base technique, log

_{9}Y becomes ( log

_{3}Y) / log

_{3}9.

Thus log

_{3}X + log

_{9}Y = log

_{3}X + (log

_{3}Y )/ log

_{3}9 = log

_{3}X + (log

_{3}Y) / 2

NOTE: log

_{3}9 = log

_{3}3

^{2}= 2 log

_{3}3 = 2(1) using Power Law of Logarithm.

log

_{3}X + log

_{9}Y = log

_{3}X + log

_{3}Y

^{(1/2)}= log

_{3}(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

_{x}7 / log

_{x}3 +2 log

_{2}7 / log

_{2}3

Answer: 3 log

_{3}7. You got it?

:-)

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