Solving logarithmic and exponential equations

To work with logarithmic equations, you need to remember the laws of logarithms:

curriculum-key-fact
  • \({\log _a}a = 1\) (since \({a^1} = a\)) so \({\log _7}7 = 1\)
  • \({\log _a}1 = 0\) (since \({a^0} = 1\)) so \({\log _{20}}1 = 0\)
  • \({\log _a}p + {\log _a}q = {\log _a}pq\) so \({\log _3}2 + {\log _3}5 = {\log _3}10\)
  • \({\log _a}p - {\log _a}q = {\log _a}\frac{p}{q}\) so \({\log _6}54 - {\log _6}9 = {\log _6}\frac{{54}}{9} = {\log _6}6 = 1\)
  • \({\log _a}{p^n} = n{\log _a}p\) so \({\log _2}{5^3} = 3{\log _2}5\)
Question

Solve for \(x\textgreater0\), \({\log _p}12 - {\log _p}x = {\log _p}1\)

Remember one of the laws of logs: \({\log _a}x - {\log _a}y = {\log _a}\frac{x}{y}\)

\[{\log _p}12 - {\log _p}x = {\log _p}1\]

\[{\log _p}\frac{{12}}{x} = {\log _p}1\]

\[\frac{{12}}{x} = 1\]

\[x = 12\]