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Popular >

9^3x+1=5^x-2

Solution

9^{3} x + 1 = 5^{x} - 2

Solution

x = - \frac{243 W _{- 1} ( - \frac{l n ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}} ) + ln ( 5 )}{243 ln ( 5 )}, x = - \frac{243 W _{0} ( - \frac{l n ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}} ) + ln ( 5 )}{243 ln ( 5 )}

+1

Decimal Notation

x = 5.10961 \dots, x = - 0.00274 \dots

Solution steps

9^{3} x + 1 = 5^{x} - 2

Prepare

9^{3} x + 1 = 5^{x} - 2

for Lambert form:

1 = (729 x + 3) e^{- l n (5) x}

Rewrite the equation with

(- x - \frac{1}{243}) ln (5) = u

and

x = - \frac{243 u + ln ( 5 )}{243 ln ( 5 )}

1 = (729 (- \frac{243 u + ln ( 5 )}{243 ln ( 5 )}) + 3) e^{- l n (5) (- \frac{243 u + l n ( 5 )}{243 l n ( 5 )})}

Simplify

1 = e^{\frac{243 u + l n ( 5 )}{243}} (- \frac{3 ( 243 u + ln ( 5 ) )}{ln ( 5 )} + 3)

Rewrite

1 = e^{\frac{243 u + l n ( 5 )}{243}} (- \frac{3 ( 243 u + ln ( 5 ) )}{ln ( 5 )} + 3)

in Lambert form:

\frac{e ^{\frac{243 u + l n ( 5 )}{243}} u}{e ^{\frac{l n ( 5 )}{243}}} = - \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}}

Solve

\frac{e ^{\frac{243 u + l n ( 5 )}{243}} u}{e ^{\frac{l n ( 5 )}{243}}} = - \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}} : u = W_{- 1} (- \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}}), u = W_{0} (- \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}})

u = W_{- 1} (- \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}}), u = W_{0} (- \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}})

Substitute back

u = (- x - \frac{1}{243}) ln (5),

solve for

x

Solve

(- x - \frac{1}{243}) ln (5) = W_{- 1} (- \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}}) : x = - \frac{243 W _{- 1} ( - \frac{l n ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}} ) + ln ( 5 )}{243 ln ( 5 )}

Solve

(- x - \frac{1}{243}) ln (5) = W_{0} (- \frac{ln ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}}) : x = - \frac{243 W _{0} ( - \frac{l n ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}} ) + ln ( 5 )}{243 ln ( 5 )}

x = - \frac{243 W _{- 1} ( - \frac{l n ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}} ) + ln ( 5 )}{243 ln ( 5 )}, x = - \frac{243 W _{0} ( - \frac{l n ( 5 )}{729 e ^{\frac{l n ( 5 )}{243}}} ) + ln ( 5 )}{243 ln ( 5 )}

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Frequently Asked Questions (FAQ)

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Solve for x: 9^3x+1=5^x-2
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