Identify errors in student work

6．（i）Eden，who is confused about the laws of logarithms，states that $$\left( \log _ { 5 } p \right) ^ { 2 } = \log _ { 5 } \left( p ^ { 2 } \right)$$ and \(\log _ { 5 } ( q - p ) = \log _ { 5 } q - \log _ { 5 } p\) However，there is a value of \(p\) and a value of \(q\) for which both statements are correct．
Determine these values．
（ii）（a）Let \(r \in \mathbb { R } ^ { + } , r \neq 1\) ．Prove that $$\log _ { r } A = \log _ { r ^ { 2 } } B \Rightarrow A ^ { 2 } = B$$ （b）Solve $$\log _ { 4 } \left( 3 x ^ { 3 } + 26 x ^ { 2 } + 40 x \right) = 2 + \log _ { 2 } ( x + 2 )$$

1. A student's attempt to solve the equation \(2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3\) is shown below.

$$\begin{aligned} & 2 \log _ { 2 } x - \log _ { 2 } \sqrt { x } = 3 \\ & 2 \log _ { 2 } \left( \frac { x } { \sqrt { x } } \right) = 3 \\ & 2 \log _ { 2 } ( \sqrt { x } ) = 3 \\ & \log _ { 2 } x = 3 \\ & x = 3 ^ { 2 } = 9 \end{aligned}$$ using the subtraction law for logs simplifying using the power law for logs using the definition of a log

1. Identify two errors made by this student, giving a brief explanation of each.
2. Write out the correct solution.

5. A student is asked to solve the equation $$\log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } = 1$$ The student's attempt is shown $$\begin{aligned} \log _ { 3 } x - \log _ { 3 } \sqrt { x - 2 } & = 1 \\ x - \sqrt { x - 2 } & = 3 ^ { 1 } \\ x - 3 & = \sqrt { x - 2 } \\ ( x - 3 ) ^ { 2 } & = x - 2 \\ x ^ { 2 } - 7 x + 11 & = 0 \\ x = \frac { 7 + \sqrt { 5 } } { 2 } \quad \text { or } \quad x & = \frac { 7 - \sqrt { 5 } } { 2 } \end{aligned}$$ a. Identify the error made by this student, giving a brief explanation.
b. Write out the correct solution.

5．［In this question the values of \(a , x\) ，and \(n\) are such that \(a\) and \(x\) are positive real numbers，with \(a > 1 , x \neq a , x \neq 1\) and \(n\) is an integer with \(n > 1\) ］ Sam was confused about the rules of logarithms and thought that $$\log _ { a } x ^ { n } = \left( \log _ { a } x \right) ^ { n }$$ （a）Given that \(x\) satisfies statement（1）find \(x\) in terms of \(a\) and \(n\) ． Sam also thought that $$\log _ { a } x + \log _ { a } x ^ { 2 } + \ldots + \log _ { a } x ^ { n } = \log _ { a } x + \left( \log _ { a } x \right) ^ { 2 } + \ldots + \left( \log _ { a } x \right) ^ { n }$$ （b）For \(n = 3 , x _ { 1 }\) and \(x _ { 2 } \left( x _ { 1 } > x _ { 2 } \right)\) are the two values of \(x\) that satisfy statement（2）．
（i）Find，in terms of \(a\) ，an expression for \(x _ { 1 }\) and an expression for \(x _ { 2 }\) ．
（ii）Find the exact value of \(\log _ { a } \left( \frac { x _ { 1 } } { x _ { 2 } } \right)\) ．
（c）Show that if \(\log _ { a } x\) satisfies statement（2）then $$2 \left( \log _ { a } x \right) ^ { n } - n ( n + 1 ) \log _ { a } x + \left( n ^ { 2 } + n - 2 \right) = 0$$

5 A student was asked to solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\). The student's attempt is written out below. $$\begin{aligned} & 2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0 \\ & 4 \log _ { 3 } x - 3 \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x - 2 = 0 \\ & \log _ { 3 } x = 2 \\ & x = 8 \end{aligned}$$

1. Identify the two mistakes that the student has made.
2. Solve the equation \(2 \left( \log _ { 3 } x \right) ^ { 2 } - 3 \log _ { 3 } x - 2 = 0\), giving your answers in an exact form.