What is the answer to x^3=3^x ?

The answer to x^3=3^x is x=3,x=-(3\W_{0}(-(ln(3))/3))/(ln(3))

The solution is x=3,x=-(3\W_{0}(-(ln(3))/3))/(ln(3))

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x^3=3^x

x^{3} = 3^{x}

x = 3, x = - \frac{3 W _{0} ( - \frac{l n ( 3 )}{3} )}{ln ( 3 )}

Solution steps

x^{3} = 3^{x}

Prepare

x^{3} = 3^{x}

for Lambert form:

x e^{- \frac{l n ( 3 ) x}{3}} = 1

Rewrite the equation with

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

and

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

(- \frac{3 u}{ln ( 3 )}) e^{u} = 1

Rewrite

(- \frac{3 u}{ln ( 3 )}) e^{u} = 1

in Lambert form:

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

Solve

e^{u} u = - \frac{ln ( 3 )}{3} : u = - ln (3), u = W_{0} (- \frac{ln ( 3 )}{3})

u = - ln (3), u = W_{0} (- \frac{ln ( 3 )}{3})

Substitute back

u = - \frac{ln ( 3 ) x}{3},

solve for

x

Solve

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

Solve

- \frac{ln ( 3 ) x}{3} = W_{0} (- \frac{ln ( 3 )}{3}) : x = - \frac{3 W _{0} ( - \frac{l n ( 3 )}{3} )}{ln ( 3 )}

x = 3, x = - \frac{3 W _{0} ( - \frac{l n ( 3 )}{3} )}{ln ( 3 )}

e^{l n (x)} = 4

5^{x + 3} \cdot 5^{2 x - 4} = 125

2 \cdot 3^{4 t} = 5^{- t}

4^{x + 2} = 64

2^{x} = 4^{x + 1}

What is the answer to x^3=3^x ?
The answer to x^3=3^x is x=3,x=-(3\W_{0}(-(ln(3))/3))/(ln(3))
Solve for x: x^3=3^x
The solution is x=3,x=-(3\W_{0}(-(ln(3))/3))/(ln(3))