First, let's expand the logarithms.
\(log _{5} 4+log_{4}5 = \frac{log4}{log5}+\frac{log5}{log4}\)
This because \(log_{y}x\) is equivalent to \(\frac{logx}{logy}\)
From there, you need to use a calculator to find the exact numerical value.
\(\frac{log4}{log5}+\frac{log5}{log4} = 2.02231716359\)