What is the answer to e^x-9x=0 ?

The answer to e^x-9x=0 is x=-\W_{-1}(-1/9),x=-\W_{0}(-1/9)

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e^x-9x=0

e^{x} - 9 x = 0

x = - W_{- 1} (- \frac{1}{9}), x = - W_{0} (- \frac{1}{9})

Decimal Notation

x = 3.42969 \dots, x = 0.12603 \dots

Solution steps

e^{x} - 9 x = 0

Prepare

e^{x} - 9 x = 0

for Lambert form:

1 = 9 x e^{- x}

Rewrite the equation with

- x = u

and

x = - u

1 = 9 (- u) e^{u}

Rewrite

1 = 9 (- u) e^{u}

in Lambert form:

e^{u} u = - \frac{1}{9}

Solve

e^{u} u = - \frac{1}{9} : u = W_{- 1} (- \frac{1}{9}), u = W_{0} (- \frac{1}{9})

u = W_{- 1} (- \frac{1}{9}), u = W_{0} (- \frac{1}{9})

Substitute back

u = - x,

solve for

x

Solve

- x = W_{- 1} (- \frac{1}{9}) : x = - W_{- 1} (- \frac{1}{9})

Solve

- x = W_{0} (- \frac{1}{9}) : x = - W_{0} (- \frac{1}{9})

x = - W_{- 1} (- \frac{1}{9}), x = - W_{0} (- \frac{1}{9})

8^{x} = 7

e^{- x} = 0.2

9^{x} + 2 \cdot 3^{x + 1} - 1 = 0

2^{x + 1} = 6

\frac{1}{128} = 2^{5 x + 3}

What is the answer to e^x-9x=0 ?
The answer to e^x-9x=0 is x=-\W_{-1}(-1/9),x=-\W_{0}(-1/9)