Simplify $\log(α+ b(x- t)+ k)$

Question

I am trying to simplify this natural log expression the best that I can, but I am unsure what to do in order to separate $b(x-t)$. Would it be $\log(b) + \log (x/t)$? so would the whole thing be:

$$ \log (\alpha + b) + \log (x/t) + \log k \quad?$$

Thank you!

Answer 1

No. You can't separate $log(A+B)$ into two logarithms, no matter what $A$ and $B$ are. The only things you can separate are $\log(AB)=\log(A)+\log(B)$, $\log_B(A)=\frac{\log(A)}{\log(B)}$ and $\log(A^B)=B\log(A)$, and their inverse operations

Answer 2

You can only write $$\log(\alpha+bx-bt+k)$$ it holds only $$\log(ab)=\log(a)+\log(b)$$ and $$\log\frac{a}{b}=\log(a)-\log(b)$$ for positive variables $a,b$