What is the answer to x=e^{(5x-6)/3} ?

The answer to x=e^{(5x-6)/3} is x=-(3\W_{-1}(-5/(3e^2)))/5 ,x=-(3\W_{0}(-5/(3e^2)))/5

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x=e^{(5x-6)/3}

x = e^{\frac{5 x - 6}{3}}

x = - \frac{3 W _{- 1} ( - \frac{5}{3 e ^{2}} )}{5}, x = - \frac{3 W _{0} ( - \frac{5}{3 e ^{2}} )}{5}

Decimal Notation

x = 1.40329 \dots, x = 0.18386 \dots

Solution steps

x = e^{\frac{5 x - 6}{3}}

Prepare

x = e^{\frac{5 x - 6}{3}}

for Lambert form:

x e^{- \frac{5 x - 6}{3}} = 1

Rewrite the equation with

- \frac{5}{3} x = u

and

x = - \frac{3 u}{5}

(- \frac{3 u}{5}) e^{- \frac{5 ( - \frac{3 u}{5} ) - 6}{3}} = 1

Simplify

- \frac{3 e ^{u + 2} u}{5} = 1

Rewrite

- \frac{3 e ^{u + 2} u}{5} = 1

in Lambert form:

e^{u} u = - \frac{5}{3 e ^{2}}

Solve

e^{u} u = - \frac{5}{3 e ^{2}} : u = W_{- 1} (- \frac{5}{3 e ^{2}}), u = W_{0} (- \frac{5}{3 e ^{2}})

u = W_{- 1} (- \frac{5}{3 e ^{2}}), u = W_{0} (- \frac{5}{3 e ^{2}})

Substitute back

u = - \frac{5}{3} x,

solve for

x

Solve

- \frac{5}{3} x = W_{- 1} (- \frac{5}{3 e ^{2}}) : x = - \frac{3 W _{- 1} ( - \frac{5}{3 e ^{2}} )}{5}

Solve

- \frac{5}{3} x = W_{0} (- \frac{5}{3 e ^{2}}) : x = - \frac{3 W _{0} ( - \frac{5}{3 e ^{2}} )}{5}

x = - \frac{3 W _{- 1} ( - \frac{5}{3 e ^{2}} )}{5}, x = - \frac{3 W _{0} ( - \frac{5}{3 e ^{2}} )}{5}

3 2^{- x + 22} = 6 4^{x}

2^{x} - 2^{x - 2} = 12

x, 3^{x} = 0

2^{x} - 2 - x = 0

3 \cdot (\frac{1}{2})^{x} = 12

What is the answer to x=e^{(5x-6)/3} ?
The answer to x=e^{(5x-6)/3} is x=-(3\W_{-1}(-5/(3e^2)))/5 ,x=-(3\W_{0}(-5/(3e^2)))/5