atestiguoki

2022-08-04

Solve equation, and express irrational solution in exact formas a decimal rounded to 3 decimal places.

${\mathrm{log}}_{2}{x}^{{\mathrm{log}}_{2}x}=4$

${\mathrm{log}}_{2}{x}^{{\mathrm{log}}_{2}x}=4$

Dwayne Hood

Beginner2022-08-05Added 10 answers

$\mathrm{log}{A}^{x}=x\mathrm{log}A$ APPLY THIS

we will get

${\mathrm{log}}_{2}x{\mathrm{log}}_{2}x=4$ , now it's base 2 and raise both equationsby 2

$({\mathrm{log}}_{2}x{)}^{2}=4$, take square root both sides

${\mathrm{log}}_{2}x=2$, now raise bothside by 2

x=4

we will get

${\mathrm{log}}_{2}x{\mathrm{log}}_{2}x=4$ , now it's base 2 and raise both equationsby 2

$({\mathrm{log}}_{2}x{)}^{2}=4$, take square root both sides

${\mathrm{log}}_{2}x=2$, now raise bothside by 2

x=4

imire37

Beginner2022-08-06Added 8 answers

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 $A={x}^{y}$

${\mathrm{log}}_{2}{x}^{{\mathrm{log}}_{2}x}=4$

$\Rightarrow ({\mathrm{log}}_{2}x{)}^{{\mathrm{log}}_{2}x}=(2{)}^{2}$

$\Rightarrow ({\mathrm{log}}_{2}x)\ast ({\mathrm{log}}_{2}x)=(2{)}^{2}$.............................[Applying formula(1)]

$\Rightarrow ({\mathrm{log}}_{2}x{)}^{2}=(2{)}^{2}$

$\Rightarrow {\mathrm{log}}_{2}x=2$............................[equating the exponentparts]

$\Rightarrow x=(2{)}^{2}$.............................[applying formula(2)]

= 4

Answer: x =4

(1) $\mathrm{log}{A}^{x}=x\ast \mathrm{log}A$

(2) if $lo{g}_{x}A=y$ , then $A={x}^{y}$

${\mathrm{log}}_{2}{x}^{{\mathrm{log}}_{2}x}=4$

$\Rightarrow ({\mathrm{log}}_{2}x{)}^{{\mathrm{log}}_{2}x}=(2{)}^{2}$

$\Rightarrow ({\mathrm{log}}_{2}x)\ast ({\mathrm{log}}_{2}x)=(2{)}^{2}$.............................[Applying formula(1)]

$\Rightarrow ({\mathrm{log}}_{2}x{)}^{2}=(2{)}^{2}$

$\Rightarrow {\mathrm{log}}_{2}x=2$............................[equating the exponentparts]

$\Rightarrow x=(2{)}^{2}$.............................[applying formula(2)]

= 4

Answer: x =4

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