2022-07-25

Solve the equation.
$-{3}^{2-x}=-81$

grocbyntza

Expert

${3}^{\left(2-x\right)}=81$ take a $\mathrm{ln}\left(\right)$ on both sides
$\left(2-x\right)\mathrm{ln}\left(3\right)=\mathrm{ln}\left(81\right)$
$x=\left[\mathrm{ln}\left(3\right)-\mathrm{ln}\left(81\right)\right]/\mathrm{ln}\left(3\right)$ or
$x=1-\mathrm{ln}\left(81\right)/\mathrm{ln}\left(3\right)=1-4$
x= -3
1. ${\mathrm{log}}_{b}\left({x}^{n}\right)=n{\mathrm{log}}_{b}x$.
2. ${\mathrm{log}}_{b}x={\mathrm{log}}_{a}x/{\mathrm{log}}_{a}b$. this is why $\mathrm{ln}\left(81\right)/\mathrm{ln}\left(3\right)={\mathrm{log}}_{3}\left(81\right)={\mathrm{log}}_{3}\left({3}^{4}\right)=4$

Israel Hale

Expert

$-{3}^{\left(2-x\right)}=-81$
$⇒{3}^{\left(2-x\right)}=81$
$=\left(3{\right)}^{4}$
$⇒\left(2-x\right)=4$....................................[equating the exponent parts]
$⇒x=2-4$
= -2

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