What is the answer to 10e^{-t}=10-2t ?

The answer to 10e^{-t}=10-2t is t=0,t=\W_{0}(-5/(e^5))+5

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10e^{-t}=10-2t

10 e^{- t} = 10 - 2 t

t = 0, t = W_{0} (- \frac{5}{e ^{5}}) + 5

Decimal Notation

t = 0, t = 4.96511 \dots

Solution steps

10 e^{- t} = 10 - 2 t

Prepare

10 e^{- t} = 10 - 2 t

for Lambert form:

10 = (10 - 2 t) e^{t}

Rewrite the equation with

t - 5 = u

and

t = u + 5

10 = (10 - 2 (u + 5)) e^{(u + 5)}

Simplify

10 = - 2 e^{u + 5} u

Rewrite

10 = - 2 e^{u + 5} u

in Lambert form:

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

Solve

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

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

Substitute back

u = t - 5,

solve for

t

Solve

t - 5 = - 5 : t = 0

Solve

t - 5 = W_{0} (- \frac{5}{e ^{5}}) : t = W_{0} (- \frac{5}{e ^{5}}) + 5

t = 0, t = W_{0} (- \frac{5}{e ^{5}}) + 5

2 \times 3^{x} + 5 \times 3^{x} - 3^{x} = 6

8^{9 - x} = 6 4^{x}

e^{8 x} + 8 e^{8 x} x = 0

125 = 2 5^{2 h + 1}

2^{(1 + 2 x)} = 32

What is the answer to 10e^{-t}=10-2t ?
The answer to 10e^{-t}=10-2t is t=0,t=\W_{0}(-5/(e^5))+5