given these $4$ functions:

$(1)(log(n))!$

$(2)(\frac{3}{2}{)}^n$

$(3)(log(n){)}^2$

$(4)\sqrt[\phantom{2}]{n}$

Find the runtime of each and prove that

$(1)<mtext></mtext><mi>(</mi><mn>2</mn><mi>)</mi>$ and $(3)<mtext></mtext><mi>(</mi><mn>4</mn><mi>)</mi>$