How do I simplify and calculate this inequality? log &#x2061;<!-- ⁡ --> ( x



Answered question


How do I simplify and calculate this inequality?
log ( x 3 ) > | x 1 |
I can't figure out how to go about solving this inequality, besides this one step:
3 log ( x ) > | x 1 |

Answer & Explanation

Allison Pena

Allison Pena

Beginner2022-07-09Added 14 answers

As said in the comments, you have two situations buit you can notice that for x = 1, the lhs and rhs are equal.
So consider the case where x = 1 ϵ and use Taylor expansion for the lhs at x = 1. Yous will easily see that the inequality is not satisfied. Since you noticed that for x = 1, l h s = r h s, the inequality will be satisfied for x > 1. So, you can now forget the absolute value in the rhs.
Now, compute the derivative of l h s r h s; it is equal to 3 / x 1 and so canceled for x = 3; the second derivative being negative, then this point corresponds to a maximum. For x = 3, l h s = 3 l o g ( 3 ) and r h s = 2; so, at this point, l h s > r h s. On the other side, you know that x moves faster then l o g ( x ); so there is a point which will corresponds to an x intercept. You will then need to solve the equation
log ( x 3 ) = x 1
log ( x 3 ) = x 1
which does not any simple analytical solution. If you plot the function, you will see that the solution is close to x = 6.71
So, the inequality is satisfied for 1 < x < 6.71

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