I have to solve the following optimisation problem
$$\max_{c,\ell, L} \{\ln(c) + \ln(\ell)\}$$
subject to the constraints: $c=Lw$ and $1=\ell +L$.
Is it okay to solve the problem
$$\max_L\{(\ln(Lw)+\ln(1-L)\}$$
Instead or can one not substitute constraints like this and have to use Lagrangian optimisation instead.
I am giving you both the solutions using and not using Lagrange optimization method for constrained optimization.
Method $1$:(Unconstrained optimization)
As you have correctly written the problem is to maximize the function $$\ln(Lw(1-L))$$ over $L\ge 1$ assuming $w>0$. Clearly $L^*=1/2$ is the maximizer.
Method $2$:(Lagrange's method)
The corresponding lagrangian becomes $$J(c,l,L,\lambda_1,\lambda_2)=\lambda_1(Lw-c)+\lambda_2(l+L-1)-\ln(cl)$$ By the first order necessary condition for having a minima, $$\dfrac{\partial J}{\partial c}=0,\ \dfrac{\partial J}{\partial l}=0,\ \dfrac{\partial J}{\partial L}=0\\ \Rightarrow -\lambda_1-\frac{1}{c}=0,\ \lambda_2-\frac{1}{l}=0,\ \lambda_1 w+\lambda_2=0$$ From these equations along with the constraints one can get $$\lambda_1=-\frac{2}{w},\ \lambda_2=2$$ Hence, the optimal solutions are $$c=\frac{w}{2},\ l=\frac{1}{2},\ L=\frac{1}{2}$$ To see that this corresponds to the maxima note that the Hessian matrix of the Lagrangian, at the optimal point becomes $$\begin{pmatrix} \frac{1}{c^2} & 0 & 0\\ 0 & \frac{1}{l^2} & 0\\ 0 & 0 & 1 \end{pmatrix}$$ which is easily seen to be a positive definite matrix and hence the optimal point is a minimizer of $-\ln(lc)$ and hence a maximizer of $\ln (lc)$.