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

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If possible ,please solve without using limits.

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Answer 1 · 2017-05-31T06:34:14

The domain is the interval #(2, 3)#

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)#