What is the answer to 6x= 10/3-6^{-x} ?

The answer to 6x= 10/3-6^{-x} is

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6x= 10/3-6^{-x}

6 x = \frac{10}{3} - 6^{- x}

x = \frac{W _{- 1} ( - \frac{l n ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}} )}{ln ( 6 )} + \frac{5}{9}, x = \frac{W _{0} ( - \frac{l n ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}} )}{ln ( 6 )} + \frac{5}{9}

Decimal Notation

x = - 1.36391 \dots, x = 0.48575 \dots

Solution steps

6 x = \frac{10}{3} - 6^{- x}

Prepare

6 x = \frac{10}{3} - 6^{- x}

for Lambert form:

- 1 = (6 x - \frac{10}{3}) e^{l n (6) x}

Rewrite the equation with

(x - \frac{5}{9}) ln (6) = u

and

x = \frac{u}{ln ( 6 )} + \frac{5}{9}

- 1 = (6 (\frac{u}{ln ( 6 )} + \frac{5}{9}) - \frac{10}{3}) e^{l n (6) (\frac{u}{l n ( 6 )} + \frac{5}{9})}

Simplify

- 1 = \frac{6 e ^{u + \frac{5}{9} l n (6)} u}{ln ( 6 )}

Rewrite

- 1 = \frac{6 e ^{u + \frac{5}{9} l n (6)} u}{ln ( 6 )}

in Lambert form:

\frac{e ^{\frac{9 u + 5 l n ( 6 )}{9}} u}{e ^{\frac{5 l n ( 6 )}{9}}} = - \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}}

Solve

\frac{e ^{\frac{9 u + 5 l n ( 6 )}{9}} u}{e ^{\frac{5 l n ( 6 )}{9}}} = - \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}} : u = W_{- 1} (- \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}}), u = W_{0} (- \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}})

u = W_{- 1} (- \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}}), u = W_{0} (- \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}})

Substitute back

u = (x - \frac{5}{9}) ln (6),

solve for

x

Solve

(x - \frac{5}{9}) ln (6) = W_{- 1} (- \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}}) : x = \frac{W _{- 1} ( - \frac{l n ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}} )}{ln ( 6 )} + \frac{5}{9}

Solve

(x - \frac{5}{9}) ln (6) = W_{0} (- \frac{ln ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}}) : x = \frac{W _{0} ( - \frac{l n ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}} )}{ln ( 6 )} + \frac{5}{9}

x = \frac{W _{- 1} ( - \frac{l n ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}} )}{ln ( 6 )} + \frac{5}{9}, x = \frac{W _{0} ( - \frac{l n ( 6 )}{6 e ^{\frac{5 l n ( 6 )}{9}}} )}{ln ( 6 )} + \frac{5}{9}

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