Can someone explain to me the parameters of a lognormal distribution? I'm doing some reading and this is the definition I got from DeGroot's book:

Does that mean the parameters are the same?  For example, assume X is lognormally distributed and Y is normally distributed where Y = log(X).  Is this saying that X and Y have the same mean and SDs even though they are different shaped distributions?  If not, what distribution is μ and σ referring to?
In other words, if someone says X is lognormally distributed with mean μ and SD σ, do I need to do any conversion so that the mean and SD are in normal terms?
 A: Wikipedia has a nice article on log-normal distributions: https://en.m.wikipedia.org/wiki/Log-normal_distribution. The article reveals that the log-normally distributed X and the normally distributed log(X) have different means and standard deviations. 
If X follows a log-normal distribution with parameters $\mu$ and $\sigma$, then $\mu$ and $\sigma$ represent the mean and standard deviation of the distribution of log(X), which is normal. In other words, the mean and standard deviation of the normally distributed log(X) are: 
Mean of $\log(X)=\mu$
SD of $\log(X) = \sigma$
The mean and standard deviation of the log-normally distributed X are as follows: 
Mean of X = $\exp(\mu + \sigma^2/2)$
SD of X = $\sqrt{\left[\exp\left(\sigma^2\right) - 1\right] \cdot \exp(2\mu + \sigma^2)}$
A: 
assume X is lognormally distributed and Y is normally distributed where Y = log(X)

This is where you are confused. You don't make assumptions on two distributions, one of which just happens to be the log of the other.
Instead, you start with a distribution $X$. Then you consider $\log X$. If $\log X\sim N(\mu,\sigma^2)$, then we say that the original distribution $X$ is lognormal with parameters $\mu$ and $\sigma^2$.
(And then the mean of $X$ is $\exp\left(\mu+\frac{\sigma^2}{2}\right)$, for instance, so the parameters are certainly not the same. This is also why it is better to speak of the "parameters" of a lognormal, rather than of the "mean and SD" - because it's very easy to get confused whether these refer to the actual mean or the log-mean, same for SD.)
