Rewrite the expression $4 \log x-2 \log \left(x^{2}+1\right)+2 \log (x-1)$ as a single logarithm
\[
4 \log x-2 \log \left(x^{2}+1\right)+2 \log (x-1)=
\]
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Answer
\(\boxed{\log \left(\frac{x^{4}(x-1)^{2}}{(x^{2}+1)^{2}}\right)}\)
Step 1 ：The question is asking to rewrite the given expression as a single logarithm. To do this, we can use the properties of logarithms.
Step 2 ：The properties of logarithms that we will use are: 1. \(a \log b = \log b^a\) 2. \(\log a - \log b = \log \frac{a}{b}\) 3. \(\log a + \log b = \log ab\)
Step 3 ：Using these properties, we can rewrite the given expression as a single logarithm.
Step 4 ：The expression \(4 \log x-2 \log \left(x^{2}+1\right)+2 \log (x-1)\) can be rewritten as a single logarithm as follows:
Step 5 ：\(\boxed{\log \left(\frac{x^{4}(x-1)^{2}}{(x^{2}+1)^{2}}\right)}\)
