The transcription is purported to be an exact copy, a fact we can't verify for certain. Additionally, de las Casas held a bias in favor of the Native Americans (unusual for his time), which could have affected exactly what he wrote or excluded. I'm also aware us English-speaking folk are precluded from reading the transcription verbatim, and have to rely on translations of (presumably) varying fidelity.
So about the Las Casas summary - and a summary is what he immediately admits it to be, no pretending to be an exact copy except some small parts which are usually clarified to be direct words from Columbus. There seems to be some modern (or maybe not modern, i don't know?) idea that because Las Casas summarizes Columbus log, and because Las Casas was a champion of Indians and against slavery that he painted a very bad picture of Columbus in it.
That just isn't true, and all you have to do is read the log yourself. The picture of Columbus in Las Casas' transcription log (as well as other works of Las Casas) is either friendly and verging on admiring, to defending him via shifting (or at least spreading) the blame to other spaniards, to at worst very brief and sporadic scolding him for his 'sins' (mainly engaging in slavery). Here's an excerpt from Las Casas 'Historia de las Indias' on his third voyage and slaver activity, which is probably as 'bad' as Las Casas goes on Columbus:
But Wednesday, August 8, a canoe came with 12 men to the caravel and they took them all, and brought them to the ship of the Admiral, and from them he chose six and sent the others to land. From this it appears that the Admiral did it [took the natives away] without scruple as he did many other times in the first navigation, it not appearing to him that it was an injustice and an offence against God and his neighbor to take free men against their will, separating fathers from their sons and wives from their husbands and who, according to natural law were married, and these could not be taken without sin and perhaps a mortal sin and the Admiral was the responsible cause — and there was the further circumstance that these people came to the ships under tacit security and promised confidence which should have been observed toward them; and this is a scandal not only to the Christians there, but to those in all the earth and to whomever should hear of this."
Source. And if you are worried that english translation is biased, I'll let you know that John Boyd Thacher who translated and wrote this work from 1903 was a massive fanboy (for the lack of better word for it) of Columbus, as visible in his treatment of this segment as he actually pushed this segment into a footnote, and added his own thoughts trying to minimize and justify Columbus action.
To go back to Las Casas and first voyage, when Columbus took some natives as slaves similar to above, the journal lacks the same subjective thoughts like for the third one above, but instead Las Casas quotes directly from the original log Columbus own words (which is a rare occurrence in Las Casas summary, reserved for parts he deemed important):
They should be good servants and intelligent, for I observed that they quickly took in what was said to them, and I believe that they would easily be made Christians, as it appeared to me that they had no religion. I, our Lord being pleased, will take hence, at the time of my departure, six natives for your Highnesses, that they may learn to speak. I saw no beast of any kind except parrots, on this island." The above is in the words of the Admiral.
Source. Markham, writing in 1893 is another who is generally sympathetic to Columbus' image, although not on the level of Thacher. If anything they show that freely available translations of Columbus log are done by people who were more inclined to praise Columbus than vilify him
I placed the above two excerpts as examples of how Las Casas treated Columbus in his summaries of the logs, and i don't think either are especially damning and demonizing. They are corroborated by facts: we do know Columbus brought natives back and they were certainly kidnapped / enslaved to do so, as was the practice. They are also corroborated by other works. Las Casas was actually quite close to Columbus family. He worked with his son Diego, supported Columbus case in the Spanish courts of law, which is why he had access to the log in the first place. The other work that is closely following the log is Ferdinand Columbus (other son of Columbus) own work The Life Of The Admiral Christopher Columbus By His Son, and the two works (Las Casas and Ferdinands) are cross referenced and match in many cases. Both are compared to Columbus' own letter following the journey, and everything (maybe some details) checks out. Other works from contemporaries also are compared, although those are distant and perhaps more inaccurate. There is no reason to think Las Casas work is any way deliberately distorting the facts. The potential bias against Columbus simply isn't there, if anything the opposite.
At this point you may be thinking: "But wait, if Las Casas didn't write anything that bad, why do we unilaterally agree Columbus was bad?" Because he was, and we know it because we have other sources saying it, especially if we move away from the voyages (even more especially the first one). We have Michael de Cunheo letter from the second voyage where he brags Columbus gave him a young girl as a slave, then he describes how he sexually assaulted her, then at the end how Columbus ordered some 1200 slaves taken and sent back. We have records from Columbus court trail that are pretty damning even when accounting for some that may be exeggerated. Even Las Casas report on events on Hispaniola, as tip-toing around Columbus as he could with blaming others instead still show how much horrible stuff happened during and after Columbus and his family tenure as rulers.