<s>[INST] Solving this logarithm equation? How do I solve this equation using common logarithms?
$\log x = 1-\log(x-3)$
 [/INST] Let $1 = \log 10$ then, $$\log x = \log 10 - \log (x - 3).$$ For all $x$ and $y$ and a constant $a$, one always has $$\log_a xy = \log_a x + \log_a y.\tag1$$ This is one of the most important logarithm rules. From this rule, we can express $\log\dfrac xy$. $$\log \frac xy = \log x\bigg(\frac 1y\bigg),$$ and from $(1)$, we see that $$\log \frac xy = \log x  + \log \frac 1y = \log x + \log y^{-1}.$$ Let $\log y^{-1} = d$ then $10^d = y^{-1}\Leftrightarrow \dfrac{1}{10^d} = \dfrac{1}{y^{-1}} \Leftrightarrow 10^{-d} = y$. $$\begin{align}\therefore \log y &= -d \\ \Leftrightarrow -\log y &= d.\end{align}$$ Since we asserted that $d = \log y^{-1}$ then, $$\log\frac xy = \log x - \log y.$$ Do you notice how useful the rule $(1)$ is? This implies that $$\begin{align} \log x &= \log 10 - \log (x - 3) \\ &\ \vdots \\ \Leftrightarrow x^2 - 3x - 10 &= 0. \end{align}$$ We can solve for $x$ in this trinomial using the quadratic formula, such that $$\begin{align} x &= \frac{3\pm 7}{2} \\ \Leftrightarrow x = -2 \ &\text{ or } \ x = 5.\end{align}$$ If you are not familiar with the quadratic formula, comment below.
</s>