Find the optimal portfolio and consumption strategies for the log utility in continuous-time, by solving the HJB

Question:

Find the optimal and consumption strategies for the log utility in continuous-time, by solving the HJB PDE (4.43). try to find a solution of the HJB PDE of the form V (t, x) = f(t) + g(t) log x.)
Portfolio
A portfolio is a grouping of financial assets such as stocks, bonds, commodities, currencies and cash equivalents, as well as their fund counterparts, including mutual, exchange-traded and closed funds. A portfolio can also consist of non-publicly...