What is the answer to e^{2b}=3+b ?

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

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e^{2b}=3+b

e^{2 b} = 3 + b

b = - \frac{W _{- 1} ( - \frac{2}{e ^{6}} ) + 6}{2}, b = - \frac{W _{0} ( - \frac{2}{e ^{6}} ) + 6}{2}

Solution steps

e^{2 b} = 3 + b

Prepare

e^{2 b} = 3 + b

for Lambert form:

1 = (3 + b) e^{- 2 b}

Rewrite the equation with

- 2 b - 6 = u

and

b = - \frac{u + 6}{2}

1 = (3 + (- \frac{u + 6}{2})) e^{- 2 (- \frac{u + 6}{2})}

Simplify

1 = e^{u + 6} (3 - \frac{u + 6}{2})

Rewrite

1 = e^{u + 6} (3 - \frac{u + 6}{2})

in Lambert form:

e^{u} u = - \frac{2}{e ^{6}}

Solve

e^{u} u = - \frac{2}{e ^{6}} : u = W_{- 1} (- \frac{2}{e ^{6}}), u = W_{0} (- \frac{2}{e ^{6}})

u = W_{- 1} (- \frac{2}{e ^{6}}), u = W_{0} (- \frac{2}{e ^{6}})

Substitute back

u = - 2 b - 6,

solve for

b

Solve

- 2 b - 6 = W_{- 1} (- \frac{2}{e ^{6}}) : b = - \frac{W _{- 1} ( - \frac{2}{e ^{6}} ) + 6}{2}

Solve

- 2 b - 6 = W_{0} (- \frac{2}{e ^{6}}) : b = - \frac{W _{0} ( - \frac{2}{e ^{6}} ) + 6}{2}

b = - \frac{W _{- 1} ( - \frac{2}{e ^{6}} ) + 6}{2}, b = - \frac{W _{0} ( - \frac{2}{e ^{6}} ) + 6}{2}

2^{2 x} + 4^{x - 1} + 44 = 2^{2 x + 2}

7^{3 x - 2} = 11

4^{x} = 11

4^{x} = 10

6 4^{x} = 51 2^{x - 5}

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