Question

Given
`log_(b)(R)=5.4`,
`log_(b)(Q)=1.2`, and
`log_(b)(T)=-1.4`

Evaluate
`log_(b)(\sqrt{(RQ)/(T) } )`

Answer 1

`\begin{array}{llll} \textit{logarithm of factors} \\\\ \log_a(xy)\implies \log_a(x)+\log_a(y) \end{array}~\hfill \begin{array}{llll} \textit{Logarithm of rationals} \\\\ \log_a\left( (x)/(y)\right)\implies \log_a(x)-\log_a(y) \end{array} \\\\\\ \textit{Logarithm of exponentials} \\\\ \log_a\left( x^b \right)\implies b\cdot \log_a(x) \\\\[-0.35em] \rule{34em}{0.25pt}`

`\log_b(R)=5.4\hspace{5em}\log_b(Q)=1.2\hspace{5em}\log_b(T)=-1.4 \\\\[-0.35em] ~\dotfill\\\\ \log_b\left(\sqrt{\cfrac{RQ}{T}} \right)\implies \log_b\left( ~~ \left( \cfrac{RQ}{T} \right)^{(1)/(2)} ~~ \right)\implies \cfrac{1}{2}\log_b\left( \cfrac{RQ}{T} \right) \\\\\\ \cfrac{1}{2}[\log_b(RQ)~~ - ~~\log_b(T)]\implies \cfrac{1}{2}[\log_b(R)+\log_b(Q)~~ - ~~\log_b(T)] \\\\\\ \cfrac{1}{2}[5.4+1.2 - (-1.4)]\implies \cfrac{1}{2}[5.4+1.2 +1.4] \implies \cfrac{1}{2}[8]\implies \text{\LARGE 4}`