MULTIPLICATION OF TWO LOGARITHMS

Before learning multiplication of two logarithms, we have to be aware of the parts of a logarithm.

Consider the logarithm given below.

logba

In the logarithm above, 'a' is called argument and 'b' is called base.

argumentandbase.png

You can multiply two logarithms, if one of the following two conditions is met.

Condition (i) :

The argument of the first logarithm and the base of the second logarithm must be same.

Condition (ii) :

The base of the first logarithm and the argument of the second logarithm must be same.

Let us see how the multiplication of two logarithms can be simplified in the following examples.

Example 1 :

Simplify : logab ⋅ logbc

In the above two logarithms, the argument of the first logarithm and the base of the second logarithm are same. 

So, we can simplify the multiplication of above two logarithms by removing the part circled in red color. 

Example 2 :

Simplify : logx⋅ logzx

In the above two logarithms, the base of the first logarithm and the argument of the second logarithm are same.

So, we can simplify the multiplication of above two logarithms by removing the part marked in red color.

Still don't understand what is explained above, please watch the video below for step by step live explanation.

Solved Problems

Problem 1 :

Solve for x :

logyx = 2/log10y 

Solution :

logyx = 2/log10y

Multiply both sides by log10y.

logy⋅ log10y = 2

Here, the base of the first logarithm and argument of the second logarithm are same.

Then,

log10x = 2

The above equation is in logarithmic form. Convert it to exponential form to solve for x.

x = 102

x = 100

Problem 2 :

If a = log2412, b = log3624 and c = log4836, then find the value of (1 + abc) in terms of b and c.

Solution :

1 + abc = 1 + log2412  log3624 ⋅ log4836

= 1 + log3612 ⋅ log4836

= 1 + log4812

= log4848 + log4812

= log48(48 ⋅ 12)

= log48(2 ⋅ 12)2

= 2log4824

= 2log3624 ⋅ log4836

= 2bc

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