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Popular >

20x+1=3^x

Solution

20 x + 1 = 3^{x}

Solution

x = - \frac{20 W _{- 1} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}, x = - \frac{20 W _{0} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

+1

Decimal Notation

x = 4, x = 0

Solution steps

20 x + 1 = 3^{x}

Prepare

20 x + 1 = 3^{x}

for Lambert form:

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

Rewrite the equation with

(- x - \frac{1}{20}) ln (3) = u

and

x = - \frac{20 u + ln ( 3 )}{20 ln ( 3 )}

(20 (- \frac{20 u + ln ( 3 )}{20 ln ( 3 )}) + 1) e^{- l n (3) (- \frac{20 u + l n ( 3 )}{20 l n ( 3 )})} = 1

Simplify

e^{\frac{20 u + l n ( 3 )}{20}} (- \frac{20 u + ln ( 3 )}{ln ( 3 )} + 1) = 1

Rewrite

e^{\frac{20 u + l n ( 3 )}{20}} (- \frac{20 u + ln ( 3 )}{ln ( 3 )} + 1) = 1

in Lambert form:

\frac{e ^{\frac{20 u + l n ( 3 )}{20}} u}{e ^{\frac{l n ( 3 )}{20}}} = - \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}

Solve

\frac{e ^{\frac{20 u + l n ( 3 )}{20}} u}{e ^{\frac{l n ( 3 )}{20}}} = - \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} : u = W_{- 1} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}), u = W_{0} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}})

u = W_{- 1} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}), u = W_{0} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}})

Substitute back

u = (- x - \frac{1}{20}) ln (3),

solve for

x

Solve

(- x - \frac{1}{20}) ln (3) = W_{- 1} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}) : x = - \frac{20 W _{- 1} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

Solve

(- x - \frac{1}{20}) ln (3) = W_{0} (- \frac{ln ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}}) : x = - \frac{20 W _{0} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

x = - \frac{20 W _{- 1} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}, x = - \frac{20 W _{0} ( - \frac{l n ( 3 )}{20 e ^{\frac{l n ( 3 )}{20}}} ) + ln ( 3 )}{20 ln ( 3 )}

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Frequently Asked Questions (FAQ)

What is the answer to 20x+1=3^x ?
The answer to 20x+1=3^x is
Solve for x: 20x+1=3^x
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