5.3 Equations Involving Indices

METHOD:

1. Comparison of indices or base
1. If  the base are the same , when $a^x=a^y$ , then x = y
2. If  the index are the same , when $a^x=b^x$ , then a = b


2. Using common logarithm (If base and index are NOT the same)

$$\begin{array}{l}{a}^x=b\Rightarrow \lg a^x=\lg b\\ x=\frac{\lg b}{\lg a}\end{array}$$

Example 1 (Index Equation - Equal base)
Solve each of the following.
(a) ${16}^x=8$
(b) $9^x{.3}^{x-1}=81$
(c) $5^{n+1}=\frac{1}{{125}^{n-1}}$

Example 2 (Solving index equation simultaneously)
Solve the following simultaneous equations.

$2^x{.4}^{2y}=8$

$\frac{3^x}{9^y}=\frac{1}{27}$