Question

How do you condense $4\log \left( x \right)-2\log \left( \left( {{x}^{2}} \right)+1 \right)+2\log \left( x-1 \right)$?

Answer 1

Hint: To condense the expression given in the above question, we have to use the properties of the logarithm function. For this, we firstly have to use the logarithm property given as $m\log a=\log \left( {{a}^{m}} \right)$ on all of the three logarithmic terms so that the coefficients of all of the logarithm terms will become equal to one. After applying the property, the given expression will become $\log \left( {{x}^{4}} \right)+\log \left[ {{\left( \left( {{x}^{2}} \right)+1 \right)}^{-2}} \right]+\log \left[ {{\left( x-1 \right)}^{2}} \right]$. Then finally using the logarithm property $\log A+\log B=\log (AB)$ repeatedly on the logarithmic terms in the obtained expression, we will get the condensed form of the given expression.

Complete step by step solution:
Let us consider the expression given in the above question as
$\Rightarrow E=4\log \left( x \right)-2\log \left( \left( {{x}^{2}} \right)+1 \right)+2\log \left( x-1 \right)$
Applying the logarithm property $m\log a=\log \left( {{a}^{m}} \right)$ on all of the three logarithm terms, the above expression will become
$\Rightarrow E=\log \left( {{x}^{4}} \right)+\log \left[ {{\left( \left( {{x}^{2}} \right)+1 \right)}^{-2}} \right]+\log \left[ {{\left( x-1 \right)}^{2}} \right]$
Now, applying the logarithmic property $\log A+\log B=\log (AB)$ on the first two logarithmic terms, we can write the above expression as
$\begin{align}
  & \Rightarrow E=\log \left[ {{x}^{4}}{{\left( \left( {{x}^{2}} \right)+1 \right)}^{-2}} \right]+\log \left[ {{\left( x-1 \right)}^{2}} \right] \\
 & \Rightarrow E=\log \left[ \dfrac{{{x}^{4}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \right]+\log \left[ {{\left( x-1 \right)}^{2}} \right] \\
\end{align}$
Finally, we again apply the logarithm property $\log A+\log B=\log (AB)$ on the two terms of the above expression to get the condensed form as
$\Rightarrow E=\log \left[ \dfrac{{{x}^{4}}{{\left( x-1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \right]$
Hence, the logarithmic expression in the above question is condensed to $\log \left[ \dfrac{{{x}^{4}}{{\left( x-1 \right)}^{2}}}{{{\left( {{x}^{2}}+1 \right)}^{2}}} \right]$.

Note: It is necessary to get rid of all of the coefficients of the logarithmic terms so as to condense them. This is because the logarithmic property $\log A+\log B=\log (AB)$ can only be applied when the coefficients of all of the logarithmic terms are equal to one. Also, we must note that since the logarithm function is defined only for the positive arguments, the given expression will be defined under the constraint $x>1$.