Why are semi log graphs used in drug Dose response curves?
The reason is to make it easier to determine a suitable dose from the graph.
Normally, using a arithmetic scale, the response curve is a hyperbolic, with most of the information squished together in a small section of the graph. By using the log the function usually becomes somewhat sigmoid, with a relatively linear section in the transition from low-high response. This means that this dynamics in this section can be studied more carefully. 

See e.g. this example (with data from https://www.idbs.com/media/396649/dose-response-comparison-example-hr.jpg), where I have reversed the x and y-axes, and used a secondary y-scale to show the log dose (so the graph is flipped compared to a normal dose-response curve). Here, the orange line shows a semi-log dose-response curve, and the blue line shows the same data using a normal arithmetic scale. Note that the orange line (log dose) should be read against the right scale and the "normal" dose against the left scale):



Here it is obvious that the normal scale doesn't give any possibility to interpret the region where the dose is effective, while this is obvious in the log graph.

The statistical modelling of these data is to some extent a separate issue though, where probit model or logit models along with data transformations are often used to analyse the data. Note also that the "usefulness" of the semilog-plot only lies in the visual presentation, and whether it will add anything depends on the dose series used in an experiment. If the doses used are not exponential (as in this example), the semilog-plot will not add much. In the statistical test of an effect the log dose is also usually not important, and I suspect (as someone not working in pharmacology) that the semilog plot is to some extent a historical artifact from the time before computer analysis of results, when the appropriate dose was determined in  a more manual way (e.g. linear analysis of the middle portion of the semilog graph). 

Example data:

dose (conc.)    response
0.00001         0.993
0.0001          1.02
0.001           3.12
0.01            6.36
0.1             31.97
1               51.2
10              75.96
100             89
1000            97.932
10000           98.8