In a lot of sources, such as the Wikipedia page for log-normal distributions, the cumulative distribution function for log-normal distributions is denoted as
$$F_X(x)=\Phi\left(\frac{\ln x-\mu}{\sigma}\right)$$
where $\Phi$ is the cumulative distribution function of a normal distribution. That is, if $Z$ is normally distributed, its CDF is
$$\Phi(z)=\frac{1}{\sqrt{2\pi}}\int^x_{-\infty}\exp \left(-\frac{1}{2}\cdot z^2\right)dz.$$
This might be a stupid questions but I am not sure what the CDF for the log-normal distribution actually means. Is it
$$F_X(x)=\Phi\left(\frac{\ln x-\mu}{\sigma}\right) \\ =\frac{1}{\sqrt{2\pi}}\int^x_{-\infty}\exp \left(-\frac{1}{2}\cdot\left(\frac{\ln x-\mu}{\sigma}\right)^2\right)dz$$ and is there a way to derive the CDF for log-normal distribution without using its probability density function? Is the CDF of a log-normal distribution the logarithm of the CDF for normal distribution?