What is the answer to x^5=5^x ?

The answer to x^5=5^x is x=5,x=-(5\W_{0}(-(ln(5))/5))/(ln(5))

The solution is x=5,x=-(5\W_{0}(-(ln(5))/5))/(ln(5))

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x^5=5^x

x^{5} = 5^{x}

x = 5, x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

Decimal Notation

x = 5, x = 1.76492 \dots

Solution steps

x^{5} = 5^{x}

Prepare

x^{5} = 5^{x}

for Lambert form:

x e^{- \frac{l n ( 5 ) x}{5}} = 1

Rewrite the equation with

- \frac{ln ( 5 ) x}{5} = u

and

x = - \frac{5 u}{ln ( 5 )}

(- \frac{5 u}{ln ( 5 )}) e^{u} = 1

Rewrite

(- \frac{5 u}{ln ( 5 )}) e^{u} = 1

in Lambert form:

e^{u} u = - \frac{ln ( 5 )}{5}

Solve

e^{u} u = - \frac{ln ( 5 )}{5} : u = - ln (5), u = W_{0} (- \frac{ln ( 5 )}{5})

u = - ln (5), u = W_{0} (- \frac{ln ( 5 )}{5})

Substitute back

u = - \frac{ln ( 5 ) x}{5},

solve for

x

Solve

- \frac{ln ( 5 ) x}{5} = - ln (5) : x = 5

Solve

- \frac{ln ( 5 ) x}{5} = W_{0} (- \frac{ln ( 5 )}{5}) : x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

x = 5, x = - \frac{5 W _{0} ( - \frac{l n ( 5 )}{5} )}{ln ( 5 )}

4^{x + 1} = 64

2^{2 x + 1} = 4

e^{x} = - 3

(\frac{1}{4})^{4 x} = 0.25

e^{2 x} - e^{x} = 6

What is the answer to x^5=5^x ?
The answer to x^5=5^x is x=5,x=-(5\W_{0}(-(ln(5))/5))/(ln(5))
Solve for x: x^5=5^x
The solution is x=5,x=-(5\W_{0}(-(ln(5))/5))/(ln(5))