Question

Write the expression as a single logarithm, and simplify the result, if possible.

Answer 1

To simplify the logarithmic expression, we need to remember some logarithm rules. Some of the rules are:

`\begin{gathered} \log _b(x\cdot y)=\log _bx+\log _by \\ \log _b(x)/(y)=\log _bx-\log _by \end{gathered}`

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

`\begin{gathered} \log _bx+\log _by=\log _b(x\cdot y) \\ \log _354+\log _310=\log _3(54\cdot10)=\log _3540 \end{gathered}`

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.

`\begin{gathered} \log _bx-\log _by=\log _b(x)/(y) \\ \log _3540-\log _320=\log _3(540)/(20)=\log _327=3 \end{gathered}`

The single logarithm is:

`\log _3(54\cdot10)/(20)`

which then simplified to:

`\log _327`

which is equal to 3.