Koalas cannot live in the rainforest because there are no eucalyptus trees in rain forests. Koalas rely on certain types of gum trees for their food, and these trees are found mostly in dry sclerophyll forest, commonly known as bushland.
Regarding whether koalas live in the jungle: there are no jungles in Australia. There is the bush and the outback. Koalas live in eucalyptus bushland or low woodland: wherever there are sufficient eucalyptus trees of their preferred variety.
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Koalas are tree dwellers. So given the choices in the question, forest would be the closest fit as an answer. However, the more appropriate Australian terminology for forest would be "bushland" or "woodlands".
Koalas live in a range of bushland habitats, as long as there are abundant eucalyptus trees of the sort they prefer. They cannot live in the desert or rainforest, as there are no eucalyptus trees in these habitats, and they cannot live in jungles as Australia does not have "jungles".
Koalas do not live in forests.
Koalas live in eucalyptus bushland, and this is not referred to as "forests" in Australia. They are not found anywhere else.