What is the answer to (x)=3^{-x} ?

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

Solve for x: (x)=3^{-x}

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

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(x) = 3^{- x}

x = \frac{W _{0} ( ln ( 3 ) )}{ln ( 3 )}

Decimal Notation

x = 0.54780 \dots

Solution steps

(x) = 3^{- x}

Prepare

(x) = 3^{- x}

for Lambert form:

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

Rewrite the equation with

x ln (3) = u

and

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

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

Rewrite

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

in Lambert form:

e^{u} u = ln (3)

Solve

e^{u} u = ln (3) : u = W_{0} (ln (3))

u = W_{0} (ln (3))

Substitute back

u = x ln (3),

solve for

x

Solve

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

x = \frac{W _{0} ( ln ( 3 ) )}{ln ( 3 )}

x, x + y + e^{x y} = 0

2^{2 x + 1} = 3 (2^{x}) + 2

3^{x} + 16 = 9 \cdot 3^{x}

4^{v + 9} + 5 = 38

- 10 \cdot 5^{8 - 4 x} - 7 = - 12

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