Question 924370: domain and range of log(3,x+8)-5
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! x has to be greater than -8 because if x was less than or equal to -8, the log function would be invalid since you have to take the log of a positive number only in the real number system.
it appears that y can be any real number because as x approaches -8, y approaches minus infinity and as x approaches plus infinity, y approaches plus infinity.
this is really hard to see, especially on the high end of x because my graphing calculator doesn't go anywhere near as high to get an estimate of the value of y when x approaches infinity.
i resorted to finding an online calculator that would tell me the domain and range of the function.
that calculator (at least one of them) can be found at:
http://www.studygeek.org/free-math-solvers/domain-and-range-calculator/
that calculator confirmed what i suspected but wasn't smart enough, or bold enough, to emphatically state on my own.
the difficulty on the high end of x lies in the fact that the function really flattens out as your value of x gets astronomically large and almost looks like it will reach a limit and not go beyond that.
by definition, the horizontal asymptote exists if the function approaches a limit as the value of x goes to infinity.
functions with horizontal asymptotes are rational functions where there is a denominator that approaches infinity as the numberator approaches some number.
that doesn't appear to be the case with this function as there is no denominator that will approach infinity.
based on that alone, one would assume there is no horizontal asymptote.
i allowed the domain and range calculator to make the decision for me.
it says that the value of y is all real numbers from minus infinity to plus infinity.
i tried to find another calculator to confirm that the calculator i used gave me a good answer.
all the websites pointed to this same calculator by wolfram which appears to be the guru for this type of calculator.
wolfram is a highly respected math site so it's a pretty good bet that what it says is really is.
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