To simplify the logarithmic expression, we need to remember some logarithm rules. Some of the rules are:
[tex]\begin{gathered} \log _b(x\cdot y)=\log _bx+\log _by \\ \log _b\frac{x}{y}=\log _bx-\log _by \end{gathered}[/tex]
In the expression that we have, we can group it into (log₃ 54 + log₃ 10) - log₃ 20.
We can apply the logarithm product rule in this expression: log₃ 54 + log₃ 10. This becomes log₃ (54 x 10) = log₃ 540
[tex]\begin{gathered} \log _bx+\log _by=\log _b(x\cdot y) \\ \log _354+\log _310=\log _3(54\cdot10)=\log _3540 \end{gathered}[/tex]
Then, we can apply the logarithm quotient rule in this expression log₃ 540 - log₃ 20. This becomes log₃ (540/20) = log₃ 27 which is equal to 3.
[tex]\begin{gathered} \log _bx-\log _by=\log _b\frac{x}{y} \\ \log _3540-\log _320=\log _3\frac{540}{20}=\log _327=3 \end{gathered}[/tex]
The single logarithm is:
[tex]\log _3\frac{54\cdot10}{20}[/tex]
which then simplified to:
[tex]\log _327[/tex]
which is equal to 3.
