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Finding Domain for Natural Log with Exponent f(x)=ln(x^2−5x)

RidiculousName

New member
Jun 27, 2018
28
I just asked a similar question, but I got help for that one, and now I am stumped again.

I need to find the domain for \(\displaystyle f(x) = ln(x^2-5x)\)

What's confusing me is how to deal with the exponent. I can't think of a way to get around it.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,735
Okay, we require:

\(\displaystyle x^2-5x>0\)

Can you factor the expression on the LHS?
 

Country Boy

Well-known member
MHB Math Helper
Jan 30, 2018
371
A product of two numbers is positive if and only if both factors have the same sign- both negative or both negative.
 

RidiculousName

New member
Jun 27, 2018
28
Okay, we require:

\(\displaystyle x^2-5x>0\)

Can you factor the expression on the LHS?
\(\displaystyle x^2-5x>0 \) becomes \(\displaystyle x(x-5)>0\) or \(\displaystyle x^2>5x\) depending on what I do. I'm just not sure where to take it after that.
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,735
\(\displaystyle x^2-5x>0 \) becomes \(\displaystyle x(x-5)>0\) or \(\displaystyle x^2>5x\) depending on what I do. I'm just not sure where to take it after that.
Observing that \(x^2-5x=x(x-5)\) tells us that the roots are:

\(\displaystyle x\in\{0,5\}\)

Rather than testing intervals though, let's use what we know about the parabolic graphs of quadratic functions. We see the coefficient of the squared term is positive, which means the parabola opens upwards, and so, given that it has two real roots, we should expect the expression to be positive on either side of the two roots, and negative in between. Can you proceed?
 

RidiculousName

New member
Jun 27, 2018
28
Observing that \(x^2-5x=x(x-5)\) tells us that the roots are:

\(\displaystyle x\in\{0,5\}\)

Rather than testing intervals though, let's use what we know about the parabolic graphs of quadratic functions. We see the coefficient of the squared term is positive, which means the parabola opens upwards, and so, given that it has two real roots, we should expect the expression to be positive on either side of the two roots, and negative in between. Can you proceed?
So, since the coefficient of the squared root is positive, I can tell it's \(\displaystyle (\infty,0)\cup(5,\infty)\)?
 

MarkFL

Administrator
Staff member
Feb 24, 2012
13,735
So, since the coefficient of the squared root is positive, I can tell it's \(\displaystyle (\infty,0)\cup(5,\infty)\)?
I believe you mean:

\(\displaystyle (-\infty,0)\cup(5,\infty)\)

and yes, this is correct. (Yes)