I have data that is in an exponential form, however, I have been told to analyse it using least squares regression to get the equation y=mx+b before converting this to an exponential model. I was told to do this by getting the least squares linear:
$$y=mx+c$$
and then converting this to exponential form by:
$$\ln (y)= c+mx$$
- get the exp of both sides:
$$y= e^c +mx$$
For example, I currently have the equation: $$y= -0.01754x + 10.1704$$
But from here I am lost and am extremely uncertain as to how I take the exponential of both sides of the equation. I would really appreciate it if someone could please explain this to me, preferably with an example using numbers as well as a mathematical formula explanation as this helps me the most :)
You seem to have your power rules messed up. I suggest you do some basic exercises with numbers and with letters to convince yourself.
Also, if you find yourself confused about letters, try to use different letters when they are in different contexts. For example, don't just say 'replace $y$ with $\ln y$; instead, say $y = \ln u$.
$$y = mx + c$$
Now perform the change of variable $y = \ln u$, then
$$\ln u = mx + c$$
$$u = e^{mx+c} = e^{mx}\cdot e^c$$
Now, you can recognise this as an exponential model. Let $A = e^c$ and $k = m$, we have
$$u = Ae^{kx}.$$
Since your $c = 10.1704$, $m = -0.01754$, we get $A = e^{10.1704} = 26118.5$, and $k = -0.01754$, so
$$u = 26118.5 e^{-0.01754x}.$$
You can sanity-check your answer by plugging in a value of $x$ and see if you get back the correct $u=e^y$.