atestiguoki

2022-08-04

Solve equation, and express irrational solution in exact formas a decimal rounded to 3 decimal places.
${\mathrm{log}}_{2}{x}^{{\mathrm{log}}_{2}x}=4$

Dwayne Hood

$\mathrm{log}{A}^{x}=x\mathrm{log}A$ APPLY THIS
we will get
${\mathrm{log}}_{2}x{\mathrm{log}}_{2}x=4$ , now it's base 2 and raise both equationsby 2
$\left({\mathrm{log}}_{2}x{\right)}^{2}=4$, take square root both sides
${\mathrm{log}}_{2}x=2$, now raise bothside by 2
x=4

imire37

we have to use some formula to solve the question
(1) $\mathrm{log}{A}^{x}=x\ast \mathrm{log}A$
(2) if $lo{g}_{x}A=y$ , then $A={x}^{y}$
${\mathrm{log}}_{2}{x}^{{\mathrm{log}}_{2}x}=4$
$⇒\left({\mathrm{log}}_{2}x{\right)}^{{\mathrm{log}}_{2}x}=\left(2{\right)}^{2}$
$⇒\left({\mathrm{log}}_{2}x\right)\ast \left({\mathrm{log}}_{2}x\right)=\left(2{\right)}^{2}$.............................[Applying formula(1)]
$⇒\left({\mathrm{log}}_{2}x{\right)}^{2}=\left(2{\right)}^{2}$
$⇒{\mathrm{log}}_{2}x=2$............................[equating the exponentparts]
$⇒x=\left(2{\right)}^{2}$.............................[applying formula(2)]
= 4