# How do you solve 3^(x-1)=81?

Jul 15, 2016

$x = 5$

#### Explanation:

As ${3}^{x - 1} = 81$, we have

$x - 1 = {\log}_{3} 81 = {\log}_{3} {3}^{4}$

= 4log_(3)3=4×1=4

Hence $x = 4 + 1 = 5$.

Jul 15, 2016

$x = 5$

#### Explanation:

In this example, the fact that $81$ is one of the powers of $3 ,$ allows us to solve this equation using indices. (${3}^{4} = 81$)

If ${x}^{a} = {x}^{b} \text{ " rArr a = b}$

If the bases are the same then the indices are equal to each other.

${3}^{x - 1} = 81$
$\textcolor{w h i t e}{\times \times x} \downarrow$
${\textcolor{b l u e}{3}}^{\textcolor{red}{x - 1}} = {\textcolor{b l u e}{3}}^{\textcolor{red}{4}} \text{ the bases are equal}$

$\therefore \textcolor{red}{x - 1 = 4}$

$\text{ } x = 5$