$(15$ points$)$ Suppose a consumer has the following utility over goods $x$ and $z$ :

$U(q_x,q_z)=ln(0\mathrm{.}5q_x-1)+ln(q_z-2)$

Use this information to answer the following questions:

$($a$)(1$ point$)$ What is the constrained maximization problem?

$($b$)(4$ points$)$ What is the optimal choice for $(q_x,q_z)$ given income is $m,$ good $x$ has a price of $P_x$ and good $z$ has a price of $P_z?($Hint$/$Reminder: if have $f(x)=ln(g(x))$ where $f$ and $g$ are both functions of $x,$ then $\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}=\frac{1}{g(x)}\frac{\text{d}\text{e}\text{l}\text{g}}{\text{d}\text{e}\text{l}\text{x}}.)$

$($c$)(6$ points$)$ Given you answer in part $($b$)$ and focusing on good $x,$ what is the own price elasticity of demand, the cross$-$price elasticity of demand, and the income elasticity of demand? To make life easier $($i$.$e$.$ to get specific values, use $m=$$$100,P_x=$$$2,$ and $P_z=$$$4.$

$($d$)$ marks