WHAT is the domain of defination of log_4 (-log_1/2 (1+ 6/root(4)x) -2)?
1 Answer
Explanation:
I'm assuming this means
Let's start by finding the domain and range of
The log function is defined such that
Since
So,
lim_(x->0)log_(1/2)(1+6/root(4)(x)) tolim_(x->oo)log_(1/2)(1+6/root(4)(x))
lim_(x->0)log_(1/2)(oo) to(log_(1/2)(1))
-oo to 0 , not inclusive (since-oo is not a number and0 is only possible whenx=oo )
Finally, we check the outer log to see if it requires us to narrow down our domain even more.
log_4(-log_(1/2)(1+6/root(4)(x))-2)
This meets the requirements for the same log domain rule as listed above. So, the inside must be positive. Since we have already shown that
log_(1/2)(1+6/root(4)(x)) < -2
1+6/root(4)(x) < (1/2)^-2
1+6/root(4)(x) < 4
6/root(4)(x) < 3
2 < root(4)(x)
16 < x
So
Final Answer