Before learning multiplication of two logarithms, we have to be aware of the parts of a logarithm.
Consider the logarithm given below.
log_{b}a
In the logarithm above, 'a' is called argument and 'b' is called base.
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 : log_{a}b ⋅ log_{b}c
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 : log_{x}y ⋅ log_{z}x
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.
Problem 1 :
Solve for x :
log_{y}x = 2/log_{10}y
Solution :
log_{y}x = 2/log_{10}y
Multiply both sides by log_{10}y.
log_{y}x ⋅ log_{10}y = 2
Here, the base of the first logarithm and argument of the second logarithm are same.
Then,
log_{10}x = 2
The above equation is in logarithmic form. Convert it to exponential form to solve for x.
x = 10^{2}
x = 100
Problem 2 :
If a = log_{24}12, b = log_{36}24 and c = log_{48}36, then find the value of (1 + abc) in terms of b and c.
Solution :
1 + abc = 1 + log_{24}12 ⋅ log_{36}24 ⋅ log_{48}36
= 1 + log_{36}12 ⋅ log_{48}36
= 1 + log_{48}12
= log_{48}48 + log_{48}12
= log_{48}(48 ⋅ 12)
= log_{48}(2 ⋅ 12)^{2}
= 2log_{48}24
= 2log_{36}24 ⋅ log_{48}36
= 2bc
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