The $\mu$ and $\sigma^2$ parameters are the population mean and variance of the logs of the lognormal random variable with those parameters.
Your equations for them are correct - they're how the population mean and variance of the lognormal relate to the mean and variance of the log-variable.
Equating those expressions to the sample mean and variance would be a reasonable thing to do --- essentially method-of-moments.
Those equations are rather straightforward to solve.
Divide the variance by the square of the mean, you get an equation in only $\sigma^2$ (one that's easily solved).
Then once you have solved that to get an estimate of $\sigma^2$, it's simple to substitute it back into the first equation to solve for your estimate of $\mu$.