A very important comment Daryl has made. It took me years to realize this and not be confused. Linear regression means the fitting form must be linear in the unknowns (for example, $a,b,c,$ that you are trying to solve for). You can have non-linearity in the independent variable $x$. So for
you would still use linear least squares fit and yes it is still called linear. This isn't quadratic squares fit or something because $a,b,c$ are still linear. If you wanted to work with something like
or even something like
now you cannot use linear least squares fit. Now you have to use non-linear least squares fit. The second one is even linear in $x$ but nonlinear in $a,b$ so have to use non-linear least squares. So generally there are only two categories, linear and nonlinear least squares fit.
So the answer to your question is, the non-linearity in $x$ doesn't matter as long as you have linearity in the unknown coefficients so yes sine,log, exponential, any degree polynomials in $x$ are all good. You can use linear least squares on them.
But on the other hand, the example you gave $a\log(bx)+c$ is non-linear in $b$ so you have to use NLLSF on this form.
By the way, NLLSF is a weeeee bit more complicated than LLSF. Very interesting subject in its own right but will require a bit more math background. You can have things like no solutions or a huge number of finite or even infinite solution. Usually the methods aren't exact. They will be iterative meaning you cannot get an exact answer in one shot. You only get an approximation. Sometimes you need to provide a starting guess. Then you run the iteration again to get a better approximation and then again and stop whenever you think your answer is good enough. You can converge to a different solution too depending on where you started from.
LLSF is rather easy, no initial guess required, the answer is guaranteed (given enough points), the answer will be unique, the answer will be exact in one iteration, and there are fast ways of doing it too. The only "bad" thing I find annoying is that you already need to know the form of the fitting function like is it a polynomial or a trig function or a logarithm in advance. But you need that with NLLSF too.