Question

Simplify $\ln 3 - 2\ln 9 + \ln 18$.

Answer 1

Hint:
\[
1.\;\ln \left( P \right) + \ln \left( Q \right) = \ln \left( {PQ} \right) \\
2.\;\ln \left( P \right) - \ln \left( Q \right) = \ln \left( {\dfrac{P}{Q}} \right) \\
3.\;q \times \ln \left( P \right) = \ln \left( {{P^q}} \right)\]
The above given identities are some of the basic logarithmic identities. This question can be thus simplified using the application of the above given identities in the given question appropriately. Such that we have to express our given question in a form where these identities can be applied easily.


Complete step by step answer:
Given, $\ln 3 - 2\ln 9 + \ln 18.............................\left( i \right)$.
Now we have to simplify the logarithmic expressions $\ln 3 - 2\ln 9 + \ln 18$ using basic logarithmic identities.Now on observing the term $2\ln 9$ we can see that it can be expressed using power rule.Such that we know
$q \times \ln \left( P \right) = \ln \left( {{P^q}} \right) \\
\Rightarrow 2\ln 9 = \ln {9^2} = \ln 81...............\left( {ii} \right) \\ $
Substituting (ii) in (i) we can write:
$\ln 3 - 2\ln 9 + \ln 18 = \ln 3 - \ln 81 + \ln 18 \\
\Rightarrow\ln 3 - 2\ln 9 + \ln 18 = \ln 3 + \ln 18 - \ln 81...............\left( {iii} \right) \\ $
Now let’s take the term $\ln 3 + \ln 18$.Such that we have the identity \[\ln \left( P \right) + \ln \left( Q \right) = \ln \left( {PQ} \right)\], which can be applied here to simplify the term $\ln 3 + \ln 18$. So we can write:
\[\ln 3 + \ln 18 = \ln \left( {3 \times 18} \right) \\
\Rightarrow\ln 3 + \ln 18= \ln \left( {54} \right)................\left( {iv} \right) \\ \]
Now let’s substitute (iv) in (iii) and we can write:
$\ln 3 - 2\ln 9 + \ln 18 = \ln 3 + \ln 18 - \ln 81 \\
\Rightarrow\ln 3 - 2\ln 9 + \ln 18= \ln 54 - \ln 81......................\left( v \right) \\ $
Now we can simplify equation (v) using the identity,
\[\ln \left( P \right) - \ln \left( Q \right) = \ln \left( {\dfrac{P}{Q}} \right)\]
Such that we can write:
$\ln 54 - \ln 81 = \ln \left( {\dfrac{{54}}{{81}}} \right)..........................\left( {vi} \right)$
Here we can see that the equation (vi) can be simplified such that:
On simplifying we get:
 \[\ln \left( {\dfrac{{54}}{{81}}} \right) = \ln \left( {\dfrac{{2 \times 27}}{{3 \times 27}}} \right) = \ln \left( {\dfrac{2}{3}} \right).....................\left( {vii} \right)\]
Now using the identity \[\ln \left( P \right) - \ln \left( Q \right) = \ln \left( {\dfrac{P}{Q}} \right)\] we can write (vii) as:
\[\therefore\ln \left( {\dfrac{2}{3}} \right) = \ln \left( 2 \right) - \ln \left( 3 \right).............................\left( {viii} \right)\]

Therefore to simplify $\ln 3 - 2\ln 9 + \ln 18$ we get \[\ln \left( 2 \right) - \ln \left( 3 \right)\].

Note:Logarithmic properties useful for similar questions of derivatives are listed below:
$
1.\;\ln \left( {PQ} \right) = \ln \left( P \right) + \ln \left( Q \right) \\
2.\;\ln \left( {\dfrac{P}{Q}} \right) = \ln \left( P \right) - \ln \left( Q \right) \\
3.\;\ln \left( {{P^q}} \right) = q \times \ln \left( P \right) \\ $
Equations ‘1’ ‘2’ and ‘3’ are called Product Rule, Quotient Rule and Power Rule respectively.Also another basic identities which are necessary to solve logarithmic questions is $\ln 1 = 0$ where ‘a’ is any real number. Common logarithmic functions are log functions with base 10, and natural logarithmic functions are log functions with base ‘e’. Natural logarithmic functions can be represented by $\ln x\;{\text{or}}\;{\log _e}x$.