Question

How do you condense $3\ln 6+3\ln x$?

Answer 1

Hint: We are given a question based on logarithmic function which we have to condense. We will first take 3 common out as it is present in the both the terms in the given expression. Then, we will use the logarithm formula, which is, \[\ln a+\ln b=\ln ab\]. Using which we will get the expression as \[\ln 6x\]. We will then use the other logarithm formula, which is, \[\ln {{a}^{x}}=x\ln a\]. Solving further, we will get the condensed form of the given expression.

Complete step by step solution:
According to the given question, we are given an expression which is based on logarithmic function, and we have to condense the given expression. That is, we have to reduce the expression in the simplest possible way.
We have the given expression as,
$3\ln 6+3\ln x$-----(1)
We will first take 3 out as it is common in the both the terms in the equation (1), we get,
\[\Rightarrow 3(\ln 6+\ln x)\]----(2)
Now, we will use one of the logarithm property, which is,
\[\ln a+\ln b=\ln ab\]
Using the above property, we get the expression as,
\[\Rightarrow 3(\ln 6x)\]
And can be also be viewed as,
\[\Rightarrow 3\ln 6x\]-----(3)
We will now be using another logarithm property, which is, \[\ln {{a}^{x}}=x\ln a\]
\[\Rightarrow \ln {{(6x)}^{3}}\]
Solving the above expression, we get the new expression as,
\[\Rightarrow \ln 216{{x}^{3}}\]
Therefore, the condensed form of the given expression is:
\[3\ln 6+3\ln x=\ln 216{{x}^{3}}\]

Note: The commonly used logarithm properties are:
1) \[\ln a+\ln b=\ln ab\]
2) \[\ln a-\ln b=\ln \dfrac{a}{b}\]
3) \[\ln {{a}^{x}}=x\ln a\]
Also, while using these properties, care should be taken as in which form the required formula is applicable.
We can also solve by first using the property \[\ln {{a}^{x}}=x\ln a\] to express the two terms and then the property \[\ln a+\ln b=\ln ab\] to get the final answer.