I understand that more lives would be lost if we didn't and that both the cities were producers of military weapons but why didn't we bomb a military base? It seems like it would've given the Japanese the same idea and also destroyed more of their military and less civilians.
In order to decide the most adequate location to drop the atomic bombs the "Target Committee" was set up to establish ideal locations for the dropping of the bomb. Their finding are recorded in the document titled: Minutes of the second meeting of the Target Committee, Los Alamos, May 10-11, 1945.
In this particular document they brought up two points directly relating to your question. In regard to the bombs use against solely military targets there were two objections:
A) "It was agreed that psychological factors in the target selection were of great importance".
The Targetting Committee recognised that Japanese will to fight was enormous. In order to break this will to fight by "obtaining the greatest psychological effect against Japan".
B) "It was agreed that for the initial use of the weapon any small and strictly military objective should be located in a much larger area subject to blast damage in order to avoid undue risks of the weapon being lost due to bad placing of the bomb."
The committee determined that they could not risk missing with the bomb in an attempt to hit an isolated military target. They instead decided to bomb an area which was very unlikely to miss and was strategically important militarily.
The Targeting Committee located a number of targets that fit this criteria:
a) Kyoto
b) Hiroshima
c) Yokohama
d) Kokura Arsenal
Hiroshima was eventually chosen under the following pretence:
"This is an important army depot and port of embarkation in the middle of an urban industrial area. It is a good radar target and it is such a size that a large part of the city could be extensively damaged. There are adjacent hills which are likely to produce a focussing effect which would considerably increase the blast damage. Due to rivers it is not a good incendiary target. (Classified as an AA Target)"
Although it does not specifically mention Nagasaki, I would imagine that it was chosen because it fit the aforementioned criteria as bombing targets.
In summary, the bomb was very valuable and could not afford to be dropped and not have a massive impact. It also served dual purposes. Both to damage Japanese industry and ability to continue the war, but to have a massive psychological impact that would destroy the Japanese will to fight. We know today it succeeded at least in the latter.
I recommend reading the document yourself so you can make your own conclusions. It can be accessed here: