# Probability density function of a log-normal distribution only knowing the geometric mean and geometric standard deviation

I want to plot the curve of a log-normal distribution of which I only know that the mean is $10^{-7}$ and the geometric standard deviation is $1.2$.

The probability density function of a log-normal distribution is given by:

$$f(x) = \frac{1}{x}\cdot\frac{1}{\sigma\sqrt{2\pi}}\exp\left(-\frac{(\ln x - \mu)^2}{2\sigma^2}\right)$$

However here $\sigma$ is the (arithmetic) standard deviation and $\mu$ is the arithmetic mean. The geometric mean and standard deviation are defined as

$$GM = e^{\mu_l}\quad\mathrm{where}\quad\mu_l = \frac{\sum_{i=1}^N\ln(x_i)}{N}$$

and

$$GSD = e^{\sigma_l}\quad\mathrm{where}\quad\sigma_l=\sqrt{\frac{\sum_{i=1}^N\left[\ln(x_i)-\mu_l\right]^2}{N}}$$

How can I calculate the probability density function $f(x)$ if I only know the geometric mean and standard deviation?

With the help of the answer to this question I was also able to calculate the probability density function with GM and GSD. Actually the answer is more or less in the question, but what I didn't know is that

$$\mu_l = \mu$$

and

$$\sigma_l = \sigma.$$

With that knowledge $\mu$ and $\sigma$ can easily be calculated by:

$$\mu = \ln(GM)$$

and

$$\sigma = \ln(GSD)$$

and then inserted into the probability density function, leading for e.g. $GM = 1$ and $GSD \approx 1.284$ to