What is the answer to ((t+1))/((e^t))=0.5 ?

The answer to ((t+1))/((e^t))=0.5 is t=-\W_{-1}(-(0.5)/e)-1,t=-\W_{0}(-(0.5)/e)-1

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((t+1))/((e^t))=0.5

\frac{( t + 1 )}{( e ^{t} )} = 0.5

t = - W_{- 1} (- \frac{0.5}{e}) - 1, t = - W_{0} (- \frac{0.5}{e}) - 1

Decimal Notation

t = 1.67834 \dots, t = - 0.76803 \dots

Solution steps

\frac{( t + 1 )}{( e ^{t} )} = 0.5

Prepare

\frac{( t + 1 )}{( e ^{t} )} = 0.5

for Lambert form:

(t + 1) e^{- t} = 0.5

Rewrite the equation with

- t - 1 = u

and

t = - u - 1

((- u - 1) + 1) e^{- (- u - 1)} = 0.5

Simplify

- e^{u + 1} u = 0.5

Rewrite

- e^{u + 1} u = 0.5

in Lambert form:

e^{u} u = - \frac{0.5}{e}

Solve

e^{u} u = - \frac{0.5}{e} : u = W_{- 1} (- \frac{0.5}{e}), u = W_{0} (- \frac{0.5}{e})

u = W_{- 1} (- \frac{0.5}{e}), u = W_{0} (- \frac{0.5}{e})

Substitute back

u = - t - 1,

solve for

t

Solve

- t - 1 = W_{- 1} (- \frac{0.5}{e}) : t = - W_{- 1} (- \frac{0.5}{e}) - 1

Solve

- t - 1 = W_{0} (- \frac{0.5}{e}) : t = - W_{0} (- \frac{0.5}{e}) - 1

t = - W_{- 1} (- \frac{0.5}{e}) - 1, t = - W_{0} (- \frac{0.5}{e}) - 1

Verify Solutions:

t = - W_{- 1} (- \frac{0.5}{e}) - 1

True

, t = - W_{0} (- \frac{0.5}{e}) - 1

True

The solutions are

t = - W_{- 1} (- \frac{0.5}{e}) - 1, t = - W_{0} (- \frac{0.5}{e}) - 1

Verify Solutions:

t = - W_{- 1} (- \frac{0.5}{e}) - 1

True

, t = - W_{0} (- \frac{0.5}{e}) - 1

True

The solutions are

t = - W_{- 1} (- \frac{0.5}{e}) - 1, t = - W_{0} (- \frac{0.5}{e}) - 1

\frac{4}{3} = 2^{x}

\frac{1}{( 2 5 ^{x} )} = 12 5^{3 x + 2}

2^{x} = \frac{1}{1024}

2^{x} = \frac{4}{9}

2^{x} \cdot 4^{x} = 8^{\frac{1}{3}}

What is the answer to ((t+1))/((e^t))=0.5 ?
The answer to ((t+1))/((e^t))=0.5 is t=-\W_{-1}(-(0.5)/e)-1,t=-\W_{0}(-(0.5)/e)-1