Determine the values of a and n leading to the optimal curve fit
You haven't defined what you want to optimize about your curve fit, so you can't really answer this question. However it appears your instructors want to steer you toward one fairly common method.

You have the formula 

$$I = aV^n$$

You can't use the usual linear regression technique to find the parameters \$a\$ and \$n\$ because this is not a linear formula.

But if you take the log of both sides you get

$$\ln I = n \ln{aV}= n\left(\ln a + \ln V \right)$$

By defining \$A = n\ln a\$ and \$B = n\$, you can then re-write this as

$$ \ln I = A + B \ln V $$

And you have something you can apply the usual linear regression method to.

Of course this solution won't be optimized for minimum mean squared error between \$I\$ and the estimated \$I\$, but for minimum mean squared error between \$\ln I\$ and the estimated \$\ln I\$.

Note: I have been sloppy about taking the logarithm of dimensional quantities (quantities with units) here. I'm basically treating \$I\$ as "current in milliamps", a dimensionless quantity, and \$V\$ as "voltage in volts", another dimensionless quantity. You could choose other scaling factors than 1 mA for current and 1 V for voltages, but you should end up with the same result in the end.