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

The answer to 3^x=x^3 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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3^x=x^3

3^{x} = x^{3}

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

Decimal Notation

x = 3, x = 2.47805 \dots

Solution steps

3^{x} = x^{3}

Prepare

3^{x} = x^{3}

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Rewrite

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

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 )}

8 1^{5 x - 2} = 2 7^{x + 4}

4^{x - 1} = 5

a, (x^{a})^{b} = b x^{a}

10000 = 40000 e^{- 0.06 t}

1 1^{x} = 1 1^{4} \cdot 1 1^{5}

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