Write the logarithmic equation for :

$x = ab \sqrt{\dfrac{a - b}{a + b}}$

Given,

$\Rightarrow x = ab \sqrt{\dfrac{a - b}{a + b}} \\[1em] \text{Taking log on Both sides,} \\[1em] \Rightarrow \log \space x = \log \space \Big(ab \sqrt{\dfrac{a - b}{a + b}}\Big) \\[1em] \Rightarrow \log \space x = \log \space ab + \log \space\sqrt{\dfrac{a - b}{a + b}} \\[1em] \Rightarrow \log \space x = \log \space a + \log \space b + \log \space \Big({\dfrac{a - b}{a + b}}\Big)^{\dfrac{1}{2}} \\[1em] \Rightarrow \log \space x = \log \space a + \log \space b + \dfrac{1}{2} \log \space {\dfrac{a - b}{a + b}} \\[1em] \Rightarrow \log \space x = \log \space a + \log \space b + \dfrac{1}{2} [\log \space ({a - b}) - \log \space ({a + b})] \\[1em]$

Hence, logarithmic equation is

$\log \space x = \log \space a + \log \space b + \dfrac{1}{2} [\log \space ({a - b}) - \log \space ({a + b})]$