Simplify $(3\log x) - (2\log x)$ How to simplify $(3\log x) - (2\log x)$? Would this become $(\log x )^ {\frac{3}{2}}$ or would this be just $3\log x-2\log x =\log x$? If so how to get $\log x$?
I was given this question: solve for $x$ if $\log x + \log x^2 +...+ \log x^n =n(n+1)$. But, the answer to my main question will also be enough. Thank you for trying!

$3\log x - 2\log x = \log x$, just like $3y-2y=y$ no matter what $y$ is equal to.
Alternatively, you can get
$$3\log x - 2\log x = \log(x^3)-\log(x^2) = \log\left(\frac{x^3}{x^2}\right) = \log x$$
but you can never under any manipulation get $$3\log x - 2\log x = \log(x)^\frac{3}{2}$$ because that equality is simply not true.

That said, for your main question, use the fact that $\log(x^n) = n\cdot \log(x)$ and your equation should become much much simpler.
