Find the domain and range of each of the following functions involving logarithms. Explain your answers.
f(x)=log_{(x+5)}(x5)
g(x)=log_{(x5)}(x+5)
log(x) regardless of base is undefined (i.e. not real) for nonpositive numbers so for f(x), x must be greater than 5. There's no limit to how high x can be, but the base x+5 log of x5 is always below 1, but asymptotically approaches it on the right.
As x gets closer to 5 (from the right) f(x) will decrease without limit.
So f(x) has domain 5 < x and range f(x)<1
g(x) is undefined below x=5 because of the negative base. The domain is the same as for g(x). It's range is however g(x)>1, as it asymptotically approaches 1 on the right, from above.
However, added to these continuous portions, there could also be discontinuous portions of the domains.
A negative number raised to an odd integral power is another negative number. A negative number raised to an even integral power is a positive number. If these numbers differ by 10, it's possible to get isolated points with rational values to the left of the abovementioned limits of the domains. Even nonintegral logs could result from rational exponents for negative numbers where the denominator of the exponent is an odd number. Finding such would seem to be tricky while arranging a difference of 10 between the exponent and the resulting value.

Posted by Charlie
on 20150402 14:13:46 