Find the domain and range of each of the following functions involving logarithms. Explain your answers.
log(x) regardless of base is undefined (i.e. not real) for non-positive 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 x-5 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 above-mentioned limits of the domains. Even non-integral 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 2015-04-02 14:13:46