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Popular >

2^x=1+x

Solution

2^{x} = 1 + x

Solution

x = 1, x = 0

Solution steps

2^{x} = 1 + x

Prepare

2^{x} = 1 + x

for Lambert form:

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

Rewrite the equation with

(- x - 1) ln (2) = u

and

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

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

Simplify

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

Rewrite

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

in Lambert form:

e^{u} u = - \frac{ln ( 2 )}{2}

Solve

e^{u} u = - \frac{ln ( 2 )}{2} : u = - 2 ln (2), u = - ln (2)

u = - 2 ln (2), u = - ln (2)

Substitute back

u = (- x - 1) ln (2),

solve for

x

Solve

(- x - 1) ln (2) = - 2 ln (2) : x = 1

Solve

(- x - 1) ln (2) = - ln (2) : x = 0

x = 1, x = 0

Popular Examples

log_{10}(10^{3x})=12

lo g_{10} (1 0^{3 x}) = 12

e^{- 5 x} = 0

32^{-3x}=2^{30}

3 2^{- 3 x} = 2^{30}

3^{x + 1} = 6

(x-3)^{x+5}=(x-3)^{x^2-x-3}

(x - 3)^{x + 5} = (x - 3)^{x^{2} - x - 3}

Frequently Asked Questions (FAQ)

What is the answer to 2^x=1+x ?
The answer to 2^x=1+x is x=1,x=0
Solve for x: 2^x=1+x
The solution is x=1,x=0