Suppose that you have five consumption choices: good $x_1 \cdots x_5$. An indifference surface is the set of consumption choices with a CONSTANT utility. For example if $(x_1, \cdots,x_5) = (2,1,1,1,1)$ gives the same utility as $(x_1, \cdots, x_5) = (1,1,1,1,2)$ than these are both points on the same indifference surface. An indifference map is the set of all indifference surface for EVERY given utility.

Consider the following utility map:
$\displaystyle U = \sum_{i=1}^{5}{\ln({x_i - a_i}) }$
Where $(a_1, \cdots, a_5) = (3,6,3,7,8)$

The budget constraint gives the set of possible consumption choices with a given income. If you have an income of $768 and the price of good $x_i$ is given by $p_i$. The equation for the budget line is given by: $\displaystyle 768 = \sum_{i=1}^{5}{p_i x_i}$.

A utility maximizing combination of goods $x_1 \cdots x_5$ occurs when the surface given by the budget constraint is tangent to an indifference surface.

Find $x_1$ as a function of $p_1 \cdots p_5$

$x_1 =$
(Use p1 for $p_1$ and likewise for $p_2,p_3,p_4,p_5$.

The easiest way to solve this question is using Lagrange multiplier.
We define the Lagrange function to be:
$\displaystyle \Lambda(x_1,\cdots,x_5,\lambda) = U(x_1,\cdots,x_5) -\lambda\left(\sum_{i=1}^{5}{p_i x_i} - 768\right)$

Utility is maximized when all of the partial derivatives of the Lagrange function are equal to $0$.