How do you find the vertical, horizontal or slant asymptotes for #f(x)=log_2(x+3)#?
1 Answer
Answer:
Vertical asymptote at
Explanation:
Logarithmic functions will have vertical asymptotes at whatever
This is the only kind of asymptote a log function can have. The best explanation comes from calculus, but essentially, it comes down to this:

There can't be a horizontal asymptote because no matter how large a
#y# value you may seek, you can find an#x# value that gives you that#y# . (If you want#log_2(x+3)="1,000,000"# , then you choose#x=2^"1,000,000"3.# ) Thus, log functions have no maximum (and no horizontal asymptote). 
There can't be a slant (slope) asymptote because the slopes of the tangent lines get closer to 0 as
#x# goes to infinity. (The "instantaneous slope" of#log_2(x+3)# at#x# is#ln2/(x+3)# , and as#x# gets larger, this value gets closer to 0.) In other words, no matter how close to 0 you want your "instantaneous" slope to be, there will be an#x# value that gives you that slope. Thus, log functions have no limiting rate of increase (and no slant asymptote).
So the only asymptote we have is
graph{log(x+3)/log2 [5.47, 26.55, 5.75, 10.27]}