Question

How do you simplify $\ln 3-2\left( \ln 4+\ln 8 \right)$?

Answer 1

Hint: Now the given expression is an expression in logarithm. To simplify the expression we will first write $4={{2}^{2}}$ and $8={{2}^{3}}$ Now we will use the exponent rule to simplify the obtained expression. Now simplify the expression. Now in the obtained expression we will use the division rule of logarithm and hence obtain the required simplified expression for the given expression.

Complete step by step solution:
The given expression is in the terms of ln which is natural logarithm.
Now we know that ln is nothing but the log function with base e.
Hence we can use the properties of log to simplify the equation.
Now for logarithm we have the following properties.
Multiplication rule: ${{\log }_{a}}\left( pq \right)={{\log }_{a}}p+{{\log }_{a}}q$.
Division rule: ${{\log }_{a}}\left( \dfrac{p}{q} \right)=\log p-\log q$
Exponent rule: ${{\log }_{a}}{{\left( p \right)}^{n}}=n{{\log }_{a}}p$
Now we will use the rules above to expand the given expression.
Now first let us consider the given expression $\ln 3-2\left( \ln 4+\ln 8 \right)$
We know that $4={{2}^{2}}$ and $8={{2}^{3}}$ Hence we can also rewrite the given expression as $\Rightarrow \ln 3-2\left( \ln {{2}^{2}}+\ln {{2}^{3}} \right)$
Now we know that according to the exponent law of logarithm we have $\ln {{a}^{n}}=n\ln a$ .
Hence using this in the above expression we get,
$\begin{align}
  & \Rightarrow \ln 3-2\left( 2\ln 2+3\ln 2 \right) \\
 & \Rightarrow \ln 3-2\left( 5\ln 2 \right) \\
\end{align}$
$\Rightarrow \ln 3+10\ln 2$
Now again using the exponent rule we get,
$\Rightarrow \ln 3-\ln {{2}^{10}}$
Now we know that according to division rule $\log \left( \dfrac{p}{q} \right)=\log p-\log q$ hence we get,
$\Rightarrow \ln \left( \dfrac{3}{{{2}^{10}}} \right)$

Hence now we have the expression $\ln 3-2\left( \ln 4+\ln 8 \right)$ can be written as $\ln 3-2\left( \ln 4+\ln 8 \right)$

Note: Now in the given expression $\ln 3-2\left( \ln 4+\ln 8 \right)$ we can also open the brackets using distributive property. Then we will again take 2 to power by using the exponent rule. Now we will simplify the expression by using the multiplication rule and the division rule. Hence we get a simplified expression.