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Popular >

1+x=2^x

Solution

1 + x = 2^{x}

Solution

x = 1, x = 0

Solution steps

1 + x = 2^{x}

Prepare

1 + x = 2^{x}

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Simplify

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

Rewrite

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

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

86400=50000(1+1/5)^x

86400 = 50000 (1 + \frac{1}{5})^{x}

x^{log_{3}(x)+1}=9

x^{l o g_{3} (x) + 1} = 9

3^{(x-1)}+3^x=4

3^{(x - 1)} + 3^{x} = 4

0. 5^{x} = \frac{1}{16}

3^{(x + 2)} = 7

Frequently Asked Questions (FAQ)

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