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Popular >

a^x=x

Solution

a^{x} = x

Solution

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

Solution steps

a^{x} = x

Prepare

a^{x} = x

for Lambert form:

1 = x e^{- x l n (a)}

Rewrite the equation with

- x ln (a) = u

and

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

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

Rewrite

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

in Lambert form:

e^{u} u = - ln (a)

Solve

e^{u} u = - ln (a) : u = W (- ln (a))

u = W (- ln (a))

Substitute back

u = - x ln (a),

solve for

x

Solve

- x ln (a) = W (- ln (a)) : x = - \frac{W ( - ln ( a ) )}{ln ( a )}

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

Popular Examples

3^{m + 2} = 4

2^{2x-2}-3*2^x+8=0

2^{2 x - 2} - 3 \cdot 2^{x} + 8 = 0

e^{(x^{2} - 2 x)} = 1

6^{2x-2}=4^{3x-9}

6^{2 x - 2} = 4^{3 x - 9}

e^{3 x} = e^{6 x}

Frequently Asked Questions (FAQ)

What is the answer to a^x=x ?
The answer to a^x=x is x=-(\W(-ln(a)))/(ln(a))