Question

the domain of the function f(x) = log[x] base to |x|

kindly explain a little bit more with steps
thanks

Answer 1

log_x(x) = 1
because x^1 = x
For example, log_2(2) = 1
Now, we want log_|x|([x])
For example,
log_2.3([2.3]) = log_2.3(2) = log2/log2.3 = 0.83
for x<0 things get a bit trickier
log_|-2.3|[(-2.3)] = log_2.3(-3)
but logs of negative numbers are undefined.
so, we need x>0
Also, log_0.4([0.4]) = log_0.4(0)
but log(0) is undefined, so since [x]=0 for x<1, we need x>=1
What about log_1(1)? That's 1 because 1^1=1
So the domain of f(x) is x >= 1

Go see the graph I pointed you to last time.