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Popular >

4^2x+8=3(8^x)

Solution

4^{2} x + 8 = 3 (8^{x})

Solution

x = - \frac{2 W _{- 1} ( - \frac{9 l n ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}} ) + 3 ln ( 2 )}{6 ln ( 2 )}, x = - \frac{2 W _{0} ( - \frac{9 l n ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}} ) + 3 ln ( 2 )}{6 ln ( 2 )}

+1

Decimal Notation

x = 1, x = - 0.42204 \dots

Solution steps

4^{2} x + 8 = 3 (8^{x})

Prepare

4^{2} x + 8 = 3 (8^{x})

for Lambert form:

e^{- 3 l n (2) x} (16 x + 8) = 3

Rewrite the equation with

3 ln (2) (- x - \frac{1}{2}) = u

and

x = - \frac{2 u + 3 ln ( 2 )}{6 ln ( 2 )}

e^{- 3 l n (2) (- \frac{2 u + 3 l n ( 2 )}{6 l n ( 2 )})} (16 (- \frac{2 u + 3 ln ( 2 )}{6 ln ( 2 )}) + 8) = 3

Simplify

e^{\frac{2 u + 3 l n ( 2 )}{2}} (- \frac{8 ( 2 u + 3 ln ( 2 ) )}{3 ln ( 2 )} + 8) = 3

Rewrite

e^{\frac{2 u + 3 l n ( 2 )}{2}} (- \frac{8 ( 2 u + 3 ln ( 2 ) )}{3 ln ( 2 )} + 8) = 3

in Lambert form:

\frac{e ^{\frac{2 u + 3 l n ( 2 )}{2}} u}{e ^{\frac{3 l n ( 2 )}{2}}} = - \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}}

Solve

\frac{e ^{\frac{2 u + 3 l n ( 2 )}{2}} u}{e ^{\frac{3 l n ( 2 )}{2}}} = - \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}} : u = W_{- 1} (- \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}}), u = W_{0} (- \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}})

u = W_{- 1} (- \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}}), u = W_{0} (- \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}})

Substitute back

u = 3 ln (2) (- x - \frac{1}{2}),

solve for

x

Solve

3 ln (2) (- x - \frac{1}{2}) = W_{- 1} (- \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}}) : x = - \frac{2 W _{- 1} ( - \frac{9 l n ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}} ) + 3 ln ( 2 )}{6 ln ( 2 )}

Solve

3 ln (2) (- x - \frac{1}{2}) = W_{0} (- \frac{9 ln ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}}) : x = - \frac{2 W _{0} ( - \frac{9 l n ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}} ) + 3 ln ( 2 )}{6 ln ( 2 )}

x = - \frac{2 W _{- 1} ( - \frac{9 l n ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}} ) + 3 ln ( 2 )}{6 ln ( 2 )}, x = - \frac{2 W _{0} ( - \frac{9 l n ( 2 )}{16 e ^{\frac{3 l n ( 2 )}{2}}} ) + 3 ln ( 2 )}{6 ln ( 2 )}

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Frequently Asked Questions (FAQ)

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