What is the answer to e^x= x/2+3 ?

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

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e^x= x/2+3

e^{x} = \frac{x}{2} + 3

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

Decimal Notation

x = 1.29388 \dots, x = - 5.99501 \dots

Solution steps

e^{x} = \frac{x}{2} + 3

Prepare

e^{x} = \frac{x}{2} + 3

for Lambert form:

2 = (x + 6) e^{- x}

Rewrite the equation with

- x - 6 = u

and

x = - u - 6

2 = ((- u - 6) + 6) e^{- (- u - 6)}

Simplify

2 = - e^{u + 6} u

Rewrite

2 = - e^{u + 6} u

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 = - x - 6,

solve for

x

Solve

- x - 6 = W_{- 1} (- \frac{2}{e ^{6}}) : x = - W_{- 1} (- \frac{2}{e ^{6}}) - 6

Solve

- x - 6 = W_{0} (- \frac{2}{e ^{6}}) : x = - W_{0} (- \frac{2}{e ^{6}}) - 6

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

\frac{e ^{x} + e ^{- x}}{e ^{x} - e ^{- x}} = 6

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

5^{- 2 x} = \frac{1}{25}

e^{e^{x}} = 0

x, e^{2 x} = 0

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