If log_3⁡〖5=a〗 and log⁡5=b. Express log_4⁡3 in terms of

a and b.

I can't quite parse your notation Generally, log base 3 of 5 would be written

a = log_3(5)
If that is what you are after, then I'm not sure what b is supposed to be.
How about reposting using my notation?

In any case, 43 is not a product of factors involving 3 and 5, so I suspect something else is wrong here.