Expert: Abe Mantell - 7/20/2009

Question
What are the asymptotes of the functions f(x)=4^x" and "g(x)=log5x?" Please explain. I do not have any understanding of what f(x) and g(x) or Log5x means. Reading the book did not help.

Answer
f(x)=4^x simply means f(x), or y, is assigned a value for each x-value you can use.  So, for example, if x=1, then y=4^1=4...if x=-1, then
y=4^(-1)=1/4.  As for its asymptote...what happens to f(x), or y, when
x gets "bigger in the negative direction??"  If x=-1, x=-10, x=-100, etc.?
Let's see!
f(-1)=4^(-1)=1/4
f(-10)=4^(-10)=1/4^10
f(-100)=1/4^100
Notice th denominator gets larger and larger, thus the entire fraction
gets smaller and smaller...gets closer and closer to zero (even though
it never equals zero).  Thus, we sat the horizontal asymptote is y=0.
See?

As for g(x)=log5(x), i.e. "log base 5 of x"
log5(5)=1, since 5^1=5
log5(25)=2, since 5^2=25
log5(1/5)=-1, since 5^(-1)=1/5
So, log5(x) tells us what the power of the base (5) needs to be to get
the desired input (x).  So, if y=log5(x), we can write it in exponential form as: x=5^y.  OK?
For its asymptote...logarithmic functions have a vertical asymptote where ever the input would give log(0)...so in this case, the asymptote is x=0.

I hope this helped!

TTYL, Abe