What is the domain of defination of y= log_10 (1- log_10 (x^2 -5x +16))?

If possible ,please solve without using limits.

1 Answer
May 31, 2017

The domain is the interval (2, 3)

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 log_10(t) is well defined if and only if t > 0

Note that:

x^2-5x+16 = (x-5/2)^2+39/4 > 0

for all real values of x

So:

log_10(x^2-5x+16)

is well defined for all real values of x.

In order that log_10(1-log_10(x^2-5x+16)) be defined, it is necessary and sufficient 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 0 when x=2 or x=3 and negative in between.

So the domain is (2, 3)