logarithmic expr. 3 log_2(6) + log_2(3^2) - log_2(108)

[Guest]

How could I simplify the following?

3 log 6 base 2 + log 3^2 base 2 - log 108 base 2?

 

[galactus]

Use the change of base formula.

Example:

\(\displaystyle \L\\log_{2}(6)=\frac{log{6}}{log{2}}\)

 

[pka]

\(\displaystyle 108 = 2^2 3^3 \\

\begin{array}{rcl}
3\log _2 \left( 6 \right) + \log _2 \left( {3^2 } \right) - \log _2 \left( {108} \right) & = & \log _2 \left( {6^3 } \right) + \log _2 \left( {3^2 } \right) - \log _2 \left( {2^2 3^3 } \right) \\
& = & \log _2 \left( {2^3 3^3 } \right) + \log _2 \left( {3^2 } \right) - \log _2 \left( {2^2 3^3 } \right) \\
& = & \log _2 \left( {\frac{{\left( {2^3 3^3 } \right)\left( {3^2 } \right)}}{{\left( {2^2 3^3 } \right)}}} \right) \\
\end{array}\)

Can you finish?