What is the answer to (3^x)-54x+135=0 ?

The answer to (3^x)-54x+135=0 is

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(3^x)-54x+135=0

(3^{x}) - 54 x + 135 = 0

x = - \frac{2 W _{- 1} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}, x = - \frac{2 W _{0} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

Decimal Notation

x = 4, x = 3

Solution steps

(3^{x}) - 54 x + 135 = 0

Prepare

(3^{x}) - 54 x + 135 = 0

for Lambert form:

1 = - e^{- l n (3) x} (- 54 x + 135)

Rewrite the equation with

(- x + \frac{5}{2}) ln (3) = u

and

x = - \frac{2 u - 5 ln ( 3 )}{2 ln ( 3 )}

1 = - e^{- l n (3) (- \frac{2 u - 5 l n ( 3 )}{2 l n ( 3 )})} (- 54 (- \frac{2 u - 5 ln ( 3 )}{2 ln ( 3 )}) + 135)

Simplify

1 = - e^{\frac{2 u - 5 l n ( 3 )}{2}} (\frac{27 ( 2 u - 5 ln ( 3 ) )}{ln ( 3 )} + 135)

Rewrite

1 = - e^{\frac{2 u - 5 l n ( 3 )}{2}} (\frac{27 ( 2 u - 5 ln ( 3 ) )}{ln ( 3 )} + 135)

in Lambert form:

e^{u} u = - \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}

Solve

e^{u} u = - \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54} : u = W_{- 1} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}), u = W_{0} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54})

u = W_{- 1} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}), u = W_{0} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54})

Substitute back

u = (- x + \frac{5}{2}) ln (3),

solve for

x

Solve

(- x + \frac{5}{2}) ln (3) = W_{- 1} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}) : x = - \frac{2 W _{- 1} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

Solve

(- x + \frac{5}{2}) ln (3) = W_{0} (- \frac{e ^{\frac{5 l n ( 3 )}{2}} ln ( 3 )}{54}) : x = - \frac{2 W _{0} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

x = - \frac{2 W _{- 1} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}, x = - \frac{2 W _{0} ( - \frac{e ^{\frac{5 l n ( 3 )}{2}} l n ( 3 )}{54} ) - 5 ln ( 3 )}{2 ln ( 3 )}

8 = 2^{x}

2^{x} = 3^{- x}

x, 2^{x} = \frac{( 1 + y )}{( 1 - y )}

\frac{( 5 ^{5} \cdot 5 ^{- 4} \cdot 5 ^{x} )}{( 5 ^{12} )} = 5^{- 5}

2^{2 x} = 5^{x - 1}