You can simplify the expression as following:

$$\log(n+1)=\log(n)+\log \left(\frac{n+1}{n}\right)$$

It shouldn't be to hard to show that $\log \left(\frac{n+1}{n}\right)<\log(n)$ for $n\geq 2$.

But it doesn't necessarily help to make the process shorter.

You can simplify the expression as following:

$$\log(n+1)=\log(n)+\log \left(\frac{n+1}{n}\right)$$

It shouldn't be to hard to show that $\log \left(\frac{n+1}{n}\right)<\log(n)$ for $n\geq 2$.

But it doesn't necessarily help to make the process shorter.

However, to solve the problem, you need to rewrite something. This can be done with inequalities or with equalities. In this case, I'd personally prefer to start out with inequalities.