What is the answer to e^{x-1}-1.5x=0 ?

The answer to e^{x-1}-1.5x=0 is x=-\W_{-1}(-1/(1.5e)),x=-\W_{0}(-1/(1.5e))

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e^{x-1}-1.5x=0

e^{x - 1} - 1.5 x = 0

x = - W_{- 1} (- \frac{1}{1.5 e}), x = - W_{0} (- \frac{1}{1.5 e})

Decimal Notation

x = 2.18883 \dots, x = 0.34698 \dots

Solution steps

e^{x - 1} - 1.5 x = 0

Prepare

e^{x - 1} - 1.5 x = 0

for Lambert form:

1 = 1.5 x e^{- x + 1}

Rewrite the equation with

- x = u

and

x = - u

1 = 1.5 (- u) e^{- (- u) + 1}

Simplify

1 = - 1.5 e^{u + 1} u

Rewrite

1 = - 1.5 e^{u + 1} u

in Lambert form:

e^{u} u = - \frac{1}{1.5 e}

Solve

e^{u} u = - \frac{1}{1.5 e} : u = W_{- 1} (- \frac{1}{1.5 e}), u = W_{0} (- \frac{1}{1.5 e})

u = W_{- 1} (- \frac{1}{1.5 e}), u = W_{0} (- \frac{1}{1.5 e})

Substitute back

u = - x,

solve for

x

Solve

- x = W_{- 1} (- \frac{1}{1.5 e}) : x = - W_{- 1} (- \frac{1}{1.5 e})

Solve

- x = W_{0} (- \frac{1}{1.5 e}) : x = - W_{0} (- \frac{1}{1.5 e})

x = - W_{- 1} (- \frac{1}{1.5 e}), x = - W_{0} (- \frac{1}{1.5 e})

(\frac{1}{100})^{2 x} = 1

e^{x} - 4 e^{- x} = 3

3^{2 - x} = \frac{2 7 ^{x + 1}}{9}

3^{\frac{x ^{2} - 4 x - 5}{3 x - 1}} = 1

What is the answer to e^{x-1}-1.5x=0 ?
The answer to e^{x-1}-1.5x=0 is x=-\W_{-1}(-1/(1.5e)),x=-\W_{0}(-1/(1.5e))