What is the domain of defination of #y= log_10 (1- log_10 (x^2 -5x +16))#?
If possible ,please solve without using limits.
If possible ,please solve without using limits.
1 Answer
The domain is the interval
Explanation:
Given:
#y = log_10(1-log_10(x^2-5x+16))#
Assume that we want to deal with this as a real valued function of real numbers.
Then
Note that:
#x^2-5x+16 = (x-5/2)^2+39/4 > 0#
for all real values of
So:
#log_10(x^2-5x+16)#
is well defined for all real values of
In order that
#1 - log_10(x^2-5x+16) > 0#
Hence:
#log_10(x^2-5x+16) < 1#
Taking exponents of both sides (a monotonically increasing function) we get:
#x^2-5x+16 < 10#
That is:
#x^2-5x+6 < 0#
which factors as:
#(x-2)(x-3) < 0#
The left hand side is
So the domain is