# 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)
$\begin{array}{l}{a}^{x}=b\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}⇒\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\mathrm{lg}{a}^{x}=\mathrm{lg}b\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}x=\frac{\mathrm{lg}b}{\mathrm{lg}a}\end{array}$

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