# Solving logarithmic and exponential equations

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

• $${\log _a}a = 1$$ (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: $${\log _a}x - {\log _a}y = {\log _a}\frac{x}{y}$$

${\log _p}12 - {\log _p}x = {\log _p}1$

${\log _p}\frac{{12}}{x} = {\log _p}1$

$\frac{{12}}{x} = 1$

$x = 12$