Question

8. A friend asked you to check over their work for a practice exam problem (see below). For each step, explain what they did. Be specific by including which rules they used. Also, for each step explain what they did right and what they did wrong. If there are mistakes present do not simply say they did it wrong, or something similar, instead explain what specific mistakes were made. Then explain how they can fix their own work. Avoid starting the problem over and showing them how you would do it, but rather how to take the work they did and fix it.

Answer 1

`\log _3(\frac{25\cdot(x-7)^{(2)/(3)}}{(x+2)^2})`

Explanation:

We have the following logarithm:

`2\log _3\mleft(5\mright)-(1)/(2)\log _3\mleft(x+2\mright)^4+2\log _3\mleft(\sqrt[6]{x-7}\mright)^2`

Applying the properties we can combine so that it is in a single logarithm, just like this:

`\begin{gathered} 2\log _3(5)=\log _3(5^2)=\log _3(25) \\ (1)/(2)\log _3(x+2)^4=\log _3((x+2)^{4^{}})^{(1)/(2)}=\log _3(x+2)^2 \\ 2\log _3(\sqrt[6]{x-7})^2=\log _3((\sqrt[6]{x-7})^2)^2=\log _3(\sqrt[6]{x-7})^4=\log _3(x-7)^{(2)/(3)} \\ \log _3(25^{})-\log _3(x+2)^2+\log _3(x-7)^{(2)/(3)}=\log _3(\frac{25\cdot(x-7)^{(2)/(3)}}{(x+2)^2}) \end{gathered}`