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Popular >

2x=42^x-1

Solution

2 x = 4 2^{x} - 1

Solution

x = - \frac{2 W _{- 1} ( - \frac{l n ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}} ) + ln ( 42 )}{2 ln ( 42 )}, x = - \frac{2 W _{0} ( - \frac{l n ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}} ) + ln ( 42 )}{2 ln ( 42 )}

+1

Decimal Notation

x = 0, x = - 0.37850 \dots

Solution steps

2 x = 4 2^{x} - 1

Prepare

2 x = 4 2^{x} - 1

for Lambert form:

1 = (2 x + 1) e^{- l n (42) x}

Rewrite the equation with

(- x - \frac{1}{2}) ln (42) = u

and

x = - \frac{2 u + ln ( 42 )}{2 ln ( 42 )}

1 = (2 (- \frac{2 u + ln ( 42 )}{2 ln ( 42 )}) + 1) e^{- l n (42) (- \frac{2 u + l n ( 42 )}{2 l n ( 42 )})}

Simplify

1 = e^{\frac{2 u + l n ( 42 )}{2}} (- \frac{2 u + ln ( 42 )}{ln ( 42 )} + 1)

Rewrite

1 = e^{\frac{2 u + l n ( 42 )}{2}} (- \frac{2 u + ln ( 42 )}{ln ( 42 )} + 1)

in Lambert form:

\frac{e ^{\frac{2 u + l n ( 42 )}{2}} u}{e ^{\frac{l n ( 42 )}{2}}} = - \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}}

Solve

\frac{e ^{\frac{2 u + l n ( 42 )}{2}} u}{e ^{\frac{l n ( 42 )}{2}}} = - \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}} : u = W_{- 1} (- \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}}), u = W_{0} (- \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}})

u = W_{- 1} (- \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}}), u = W_{0} (- \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}})

Substitute back

u = (- x - \frac{1}{2}) ln (42),

solve for

x

Solve

(- x - \frac{1}{2}) ln (42) = W_{- 1} (- \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}}) : x = - \frac{2 W _{- 1} ( - \frac{l n ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}} ) + ln ( 42 )}{2 ln ( 42 )}

Solve

(- x - \frac{1}{2}) ln (42) = W_{0} (- \frac{ln ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}}) : x = - \frac{2 W _{0} ( - \frac{l n ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}} ) + ln ( 42 )}{2 ln ( 42 )}

x = - \frac{2 W _{- 1} ( - \frac{l n ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}} ) + ln ( 42 )}{2 ln ( 42 )}, x = - \frac{2 W _{0} ( - \frac{l n ( 42 )}{2 e ^{\frac{l n ( 42 )}{2}}} ) + ln ( 42 )}{2 ln ( 42 )}

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Frequently Asked Questions (FAQ)

What is the answer to 2x=42^x-1 ?
The answer to 2x=42^x-1 is
Solve for x: 2x=42^x-1
The solution is