How do you condense $4\left[ \ln z+\ln \left( z+5 \right)-2\ln \left( z-5 \right) \right]$?

Hint: Before condensing the given logarithmic expression, we first need to determine the domain for the given expression. Since the logarithm functions is defined only for the positive arguments, we will have the inequalities $z=0,z+5 > 0,z-5 > 0$. On combining these inequalities we will get the domain for the given expression. Then we can condense the given expression by using the logarithm properties $\ln A-\ln B=\ln \left( \dfrac{A}{B} \right)$ and $\ln A+\ln B=\ln \left( AB \right)$.

Complete step by step solution:
Let us write the expression given in the above question as
$\Rightarrow E=4\left[ \ln z+\ln \left( z+5 \right)-2\ln \left( z-5 \right) \right]$
We know that the logarithm function is defined for the positive arguments only. Therefore from the three logarithmic terms given above, we can write the inequalities
$\Rightarrow z>0$
And
$\begin{align}
  & \Rightarrow z+5>0 \\
 & \Rightarrow z>-5 \\
\end{align}$
Also
$\begin{align}
  & \Rightarrow z-5>0 \\
 & \Rightarrow z>5 \\
\end{align}$
Combining all of the three inequalities written above, we have $z>5$, which is the domain for the given expression. Now, we again consider the given expression
$\Rightarrow E=4\left[ \ln z+\ln \left( z+5 \right)-2\ln \left( z-5 \right) \right]$
We know that $\ln A+\ln B=\ln \left( AB \right)$. Therefore we can combine the first two logarithmic terms of the above expression as
$\Rightarrow E=4\left[ \ln \left[ z\left( z+5 \right) \right]-2\ln \left( z-5 \right) \right]$
Now, we know that $m\ln a=\ln {{a}^{m}}$. So the above expression becomes
$\Rightarrow E=4\left\{ \ln \left[ z\left( z+5 \right) \right]-\ln {{\left( z-5 \right)}^{2}} \right\}$
Now applying the logarithmic property $\ln A-\ln B=\ln \left( \dfrac{A}{B} \right)$, we get
$\Rightarrow E=4\left\{ \ln \left[ \dfrac{z\left( z+5 \right)}{{{\left( z-5 \right)}^{2}}} \right] \right\}$
Finally, we again apply the logarithm property $m\ln a=\ln {{a}^{m}}$ to write the above expression as
$\begin{align}
  & \Rightarrow E=\ln {{\left[ \dfrac{z\left( z+5 \right)}{{{\left( z-5 \right)}^{2}}} \right]}^{4}} \\
 & \Rightarrow E=\ln \left[ \dfrac{{{z}^{4}}{{\left( z+5 \right)}^{4}}}{{{\left( z-5 \right)}^{8}}} \right] \\
\end{align}$
Hence, the given expression is condensed as $\ln \left[ \dfrac{{{z}^{4}}{{\left( z+5 \right)}^{4}}}{{{\left( z-5 \right)}^{8}}} \right]$ with the condition that $z > 5$.

Note: Do not forget to determine the domain for the given expression, since the arguments to the logarithm terms are variable and not constants. We can also simplify the given expression by first multiplying $4$ in the expression $4\left[ \ln z+\ln \left( z+5 \right)-2\ln \left( z-5 \right) \right]$ and using the property $m\ln a=\ln {{a}^{m}}$ write it as \[\ln {{z}^{4}}+\ln {{\left( z+5 \right)}^{4}}-\ln {{\left( z-5 \right)}^{8}}\]. Then applying the logarithm properties $\ln A-\ln B=\ln \left( \dfrac{A}{B} \right)$ and $\ln A+\ln B=\ln \left( AB \right)$, we can simplify the given expression.