This question is rather too simple. It only raises additional questions needed for clarification, such as, is this logging managed sustainably or not? Does the term "rain forest" include tropical hardwood plantations, or only wild rain forests?
About a hundred years ago, the United States faced a crisis in its forestry industry. Logging companies of that day used a "cut-out and get-out" approach, stripping the trees and leaving denuded lands in their wake. Many at that time believed the forests were inexhaustible. A man named Gifford Pinchot knew otherwise. What did Pinchot do about it? First, he went to Germany. He knew that Germany was a much smaller land area than the US, and had been using its forest resources for much longer, yet it still had healthy forest cover. Gifford Pinchot carefully studied German forestry methods. On his return to the US, he found the time was ripe: President Roosevelt had just established the National Forest Service. Pinchot, because of his knowledge of sustainable forestry, was appointed head of the Forest Service. Today there is a National Forest named after him in Washington State.
The great tropical rain forests of the world are now facing the same crisis. The Brazilian Amazon gets the most publicity, but the same crisis is happening throughout the tropics, in South America, Africa, and tropical Asia, wherever rain forests are found. In this case, there is no visionary Roosevelt to save the day; governments, desperate to develop their economies quickly, allow unsustainable practices to continue.
Must it be this way? On the one hand, unsustainable logging and ecosystem destruction; on the other, preservation of rain forest in inviolate natural preserves - are these extremes the only options? In some instances, outright preservation makes sense, as for example, the case of watershed preserves: an area near a city or town, left in its forested state to protect the groundwater supply. Other cases of the importance of preservation would include biodiversity reserves, that is, areas where the highest diversity of species are left undisturbed, in case their genetic uniqueness is needed at a later date; and research areas, where scientists can study the workings of the ecosystem without outside disturbances to skew the results.
But the fact remains, the nations where the rain forests are located have to have a functioning economy. Brunei, for example, has been able to preserve 90% of its rainforests intact largely because its economy is almost entirely focused on offshore oil drilling. As oil depletion and concern over global climate change cause shifts to other fuels, Brunei will have to shift its economic focus. This will likely bring pressure to bear on its rainforests.
In contrast to Brunei is the nearby nation of Singapore. Singapore is a powerhouse economy with a focus on global commerce; its port is the busiest in the world. It has also long since lost most of its rain forests. Several preserves exist in the interior, especially on its highest peak, Bukit Timah, and are completely protected from exploitation, but this possible only because Singapore's other economic endeavors are so strong, and its government notoriously strict.
So where does this leave the great majority of tropical rainforest nations - those neither so specialized as Brunei, nor so wealthy as Singapore? Wood is a valuable commodity in the global marketplace, and always will be so long as there is construction, furniture, and the great quantities of paper used in business, government, and publishing, even since the Internet. Many tropical hardwoods, such as teak, ebony, and mahogany, are especially prized for beauty and durability, and others, such as balsa, have specialized properties not easily substituted.
The biggest problem in tropical rainforests is that the trees are cut, but seldom replanted. Instead of sustainable forest plantations, which could produce valuable wood in perpetuity, the cut-over areas are either abandoned to erosion, or given over to agriculture and cattle. These are managed in ways unsuited to the land, and so result in its ruin. We cannot entirely blame the tropical nations for this. Much of the beef raised on former rain forest land is sold to the wealthy nations. So long as the populations in wealthy nations maintain their addiction to large quantities of meat, rain forest destruction will continue. So long as beef is more lucrative than wood, ranching interests will be given priority over sustainable forestry.
Sustainable forestry is possible in the rain forest regions. Scientists even now are finding ways to manage tropical rain forests in sustainable ways. As Gifford Pinchot understood nearly a century ago, scientific forestry must replace haphazard and short-sighted practices. But for this to work, the governments of the countries involved must see that it is to their advantage to adopt sustainable policies. All the knowledge in the world is of no avail if it is not put to use.