Q.A lab assistant prepared a solution by adding a calculated quantity of HCl gas at $25^\circ C$ to get a solution with $[H_3O^+] = 4 \times 10^{-5}$ M. Is the solution neutral, acidic or basic?

Acid/base solutions typically span the concentration range $10^{-1}$ to $10^{-7}$ M, an inconveniently wide span. Sorensen's logarithmic pH scale expresses this compactly: $pH=-\log_{10}[H_3O^+]$, so $[H_3O^+]=10^{-pH}$; the analogous quantity for hydroxide is $pOH=-\log_{10}[OH^-]$.

pH + pOH = pKw. Adding the two definitions gives $pH+pOH=-\log_{10}([H_3O^+][OH^-])=-\log_{10}K_w=pK_w$. At $25^\circ C$, $K_w=1\times10^{-14}$, so $pK_w=14$ and $pH+pOH=14$. In a neutral solution at $25^\circ C$, both pH and pOH equal 7; acidic solutions have pH < 7, basic solutions have pH > 7. …