What is the answer to 16x+e^{-x}-16[1-ln(16)]-2=0 ?

The answer to 16x+e^{-x}-16[1-ln(16)]-2=0 is

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16x+e^{-x}-16[1-ln(16)]-2=0

16 x + e^{- x} - 16 [1 - ln (16)] - 2 = 0

x = W_{- 1} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) + \frac{1}{16} (- 64 ln (2) + 18), x = W_{0} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) + \frac{1}{16} (- 64 ln (2) + 18)

Solution steps

16 x + e^{- x} - 16 [1 - ln (16)] - 2 = 0

Prepare

16 x + e^{- x} - 16 [1 - ln (16)] - 2 = 0

for Lambert form:

1 = - e^{x} (16 x + 64 ln (2) - 18)

Rewrite the equation with

x - \frac{1}{16} (- 64 ln (2) + 18) = u

and

x = u + \frac{1}{16} (- 64 ln (2) + 18)

1 = - e^{(u + \frac{1}{16} (- 64 l n (2) + 18))} (16 (u + \frac{1}{16} (- 64 ln (2) + 18)) + 64 ln (2) - 18)

Simplify

1 = - 16 e^{u + \frac{1}{16} (- 64 l n (2) + 18)} u

Rewrite

1 = - 16 e^{u + \frac{1}{16} (- 64 l n (2) + 18)} u

in Lambert form:

\frac{e ^{\frac{8 u - 32 l n ( 2 ) + 9}{8}} u}{e ^{\frac{- 64 l n ( 2 ) + 18}{16}}} = - \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}

Solve

\frac{e ^{\frac{8 u - 32 l n ( 2 ) + 9}{8}} u}{e ^{\frac{- 64 l n ( 2 ) + 18}{16}}} = - \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}} : u = W_{- 1} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}), u = W_{0} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}})

u = W_{- 1} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}), u = W_{0} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}})

Substitute back

u = x - \frac{1}{16} (- 64 ln (2) + 18),

solve for

x

Solve

x - \frac{1}{16} (- 64 ln (2) + 18) = W_{- 1} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) : x = W_{- 1} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) + \frac{1}{16} (- 64 ln (2) + 18)

Solve

x - \frac{1}{16} (- 64 ln (2) + 18) = W_{0} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) : x = W_{0} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) + \frac{1}{16} (- 64 ln (2) + 18)

x = W_{- 1} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) + \frac{1}{16} (- 64 ln (2) + 18), x = W_{0} (- \frac{1}{16 e ^{\frac{- 64 l n ( 2 ) + 18}{16}}}) + \frac{1}{16} (- 64 ln (2) + 18)

2 = 12 \cdot e^{0.7 \cdot x}

7^{2 x + 3} \cdot 49 = 7^{4 x}

5 = (2.82)^{t}

\frac{4 e ^{3 x} - 5}{e ^{3 x} - 1} = 3

350000 = 400 \cdot (1 - e^{- 0.15 t})

What is the answer to 16x+e^{-x}-16[1-ln(16)]-2=0 ?
The answer to 16x+e^{-x}-16[1-ln(16)]-2=0 is