\( \operatorname { Ln } ( x ^ { 3 } ( 2 x - 5 ) ! ) \)

To explore the fun side of logarithms, let’s break down \( \operatorname{Ln}(x^3 (2x - 5)!) \). Remember, the natural logarithm has some cool properties! You can say \( \operatorname{Ln}(ab) = \operatorname{Ln}(a) + \operatorname{Ln}(b) \). Here, you could write it as \( \operatorname{Ln}(x^3) + \operatorname{Ln}((2x - 5)!) \), leading to \( 3 \operatorname{Ln}(x) + \operatorname{Ln}((2x - 5)!) \). Now that’s more fun to work with! Now, if we wanted to toss in some common mistakes, a classic one is forgetting that logarithms can’t handle negative arguments. In our scenario, ensure that \(2x - 5 > 0\) before diving in, meaning \(x > 2.5\). If \(x\) dips below this threshold, we might end up with a log of a negative number or zero, which is a big no-no in the world of logarithms!