[tex]log_6{234}- log_6{6,5} +log_16 log_3{9}[/tex]

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To simplify the given expression:

[ \log_6{234} - \log6{6.5} + \log{16}{\log_3{9}} ]

First, use the properties of logarithms to simplify each term individually:

[ \log_6{234} = \log_6{(6 \times 39)} = \log_6{6} + \log_6{39} = 1 + \log_6{39} ]

[ \log_6{6.5} = \log_6{\frac{13}{2}} = \log_6{13} - \log_6{2} ]

[ \log_{16}{\log_3{9}} = \frac{\log3{9}}{\log{16}{e}} = \frac{\log_3{3^2}}{\frac{\log_2{e}}{\log_2{16}}} = 2 \times \log_2{16} = 2 \times 4 = 8 ]

Now substitute these simplified terms back into the original expression:

[ 1 + \log_6{39} - (\log_6{13} - \log_6{2}) + 8 ]

[ 1 + \log_6{39} - \log_6{13} + \log_6{2} + 8 ]

Therefore, the final simplified expression is:

[ 9 + \log_6{\frac{39}{13} \times 2} ]

[ 9 + \log_6{6} ]

[ 9 + 1 = 10 ]

Therefore, [ \log_6{234} - \log6{6.5} + \log{16}{\log_3{9}} = 10 ]