I have that $$z = φexp(−γx)$$
The question states: By taking an appropriate transformation of the experimental response, fit a simple linear regression model and provide estimates of φ and γ.
I have done the following: $$log(z) = log(φ) - γx$$ $$log(z) = β0 + β1xi$$
So, $$β0 = log(φ)$$ and $$β1 = -γ$$
Is it as simple as working out the value of β0 and then taking the expontential of it to get φ, and working out the value of β1 and sticking a minus sign in front of it to get γ?
Then if I were to do a confidence interval for φ and γ would I use the new numbers I've worked out, or would I work out the confidence interval β0 and β1 and then do the same transformations?
Yes, it is that simple. If your original measurements are $(x_1, z_1), (x_2,z_2), \ldots, (x_n, z_n)$, you can fit a simple linear regression model to $(x_1, \log z_1), (x_2, \log z_2),\ldots, (x_n,\log z_n)$. You then rearrange the coefficients of your linear regression to read off estimates for your original parameters $\gamma$ and $\phi$, as you've done.