find the domain of root of a logarithmic function I'm a little confused about this question since o

Grimanijd 2022-07-06 Answered
find the domain of root of a logarithmic function
I'm a little confused about this question since output of a logarithmic function varies from to .I should find the domain of this function: y = log x ( 10 x 2 ) . How can I find the interval that makes log x ( 10 x 2 ) greater than zero?
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Answers (2)

verzaadtwr
Answered 2022-07-07 Author has 17 answers
Recall for a > 0 , a 1 , b > 0
log a b = ln b ln a
Thus we have
f ( x ) = log x ( 10 x 2 ) = ln ( 10 x 2 ) ln x
Then f ( x ) 0 if and only if 10 x 2 1 , x > 1 or 0 < 10 x 2 1 , 0 < x < 1. Since the later case cannot happen, then we must have 10 x 2 1 and x > 1, which gives 1 < x 3

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Joel French
Answered 2022-07-08 Author has 10 answers
First of all, I would suggest to write your log in a fixed basis
log x ( 10 x 2 ) = ln ( 10 x 2 ) ln ( x ) .
This changes your question to
ln ( 10 x 2 ) ln ( x ) > 0
or equivalently:
ln ( x ) ln ( 10 x 2 ) > 0.
Now, you have to find all roots of 10 x 2 which gives you the intervals where ln ( 10 x 2 ) > 0 and then you are almost finished.

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