x-3-3-x-find-x-

Question Number 17742 by tawa tawa last updated on 10/Jul/17

x^{3} = 3^{x}, find x .

Commented by 1kanika# last updated on 10/Jul/17

x = 3

Answered by mrW1 last updated on 10/Jul/17

x = 3^{\frac{x}{3}} = e^{\frac{x}{3} \ln 3}

{xe}^{- \frac{x}{3} \ln 3} = 1

(- \frac{x}{3} \ln 3) e^{- \frac{x}{3} \ln 3} = - \frac{\ln 3}{3}

- \frac{x}{3} \ln 3 = W (- \frac{\ln 3}{3})

\Rightarrow x = - \frac{3}{\ln 3} W (- \frac{\ln 3}{3}) = {\begin{cases} 2.4 \dots \\ 3 \end{cases}

Commented by tawa tawa last updated on 10/Jul/17

God bless you sir . i understand it . i really appreciate .

Commented by mrW1 last updated on 10/Jul/17

solve x^{2} = 3^{x}

Commented by tawa tawa last updated on 10/Jul/17

x^{2} = 3^{x}

3^{\frac{x}{2}} = x

e^{\frac{x}{2} \ln 3} = x

1 = {xe}^{- \frac{xln 3}{2}}

- \frac{\ln 3}{2} = (- \frac{xln 3}{2}) e^{(- \frac{xln 3}{2})}

- \frac{xln 3}{2} = {We}^{(- \frac{\ln 3}{2})}

x = \frac{- {2 We}^{(- \frac{\ln 3}{2})}}{\ln 3}

x = \frac{- {2 We}^{(\frac{\ln 3}{2})}}{\ln 3}

x = \frac{- 2 \times 0.376839}{1.09861}

x = \frac{- 0.753674}{1.09861}

x = - 0.6860

Commented by mrW1 last updated on 10/Jul/17

x^{2} = 3^{x}

\Rightarrow \pm 3^{\frac{x}{2}} = x

\dots \dots

totally 3 solutions

Commented by tawa tawa last updated on 10/Jul/17

ohh, i see \dots . it almost affect the last answer

Commented by tawa tawa last updated on 10/Jul/17

Sir . . . . x^{3} has two solutions

x^{2} has three solutions \dots

or it depends on the question .

Commented by tawa tawa last updated on 10/Jul/17

Show me sir .

Commented by mrW1 last updated on 10/Jul/17

if x^{3} = 3^{x}

since 3^{x} > 0

\Rightarrow x^{3} > 0

\Rightarrow x > 0

\Rightarrow x^{3} = 3^{x} \equiv x = 3^{\frac{x}{3}}

if x^{2} = 3^{x}

x can also be negative,

\Rightarrow x^{2} = 3^{x} \equiv x = \pm 3^{\frac{x}{2}}

just like

x^{3} = 8

\Rightarrow x = {(8)}^{\frac{1}{3}} = 2

x^{3} = - 8

\Rightarrow x = {(- 8)}^{\frac{1}{3}} = - 2

but

x^{2} = 4

\Rightarrow x = \pm {(4)}^{\frac{1}{2}} = \pm 2

Commented by tawa tawa last updated on 10/Jul/17

Now, i understand sir . God bless you

Commented by mrW1 last updated on 10/Jul/17

solution for x^{4} = 3^{x}

x = \pm 3^{\frac{x}{4}}

x = \pm e^{\frac{x}{4} \ln 3}

{xe}^{- \frac{x}{4} \ln 3} = \pm 1

(- \frac{x}{4} \ln 3) e^{- \frac{x}{4} \ln 3} = \pm \frac{\ln 3}{4}

- \frac{x}{4} \ln 3 = W (\pm \frac{\ln 3}{4})

\Rightarrow x = - \frac{4}{\ln 3} W (\pm \frac{\ln 3}{4})

x = {\begin{cases} - \frac{4}{\ln 3} W (\frac{\ln 3}{4}) = - 0.80225 \\ - \frac{4}{\ln 3} W (- \frac{\ln 3}{4}) = {\begin{cases} 1.51687 \\ 7.17476 \end{cases} \end{cases}

Commented by tawa tawa last updated on 10/Jul/17

I really appreciate your effort sir . God bless you sir .