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Popular >

x+1=2^x

Solution

x + 1 = 2^{x}

Solution

x = 1, x = 0

Solution steps

x + 1 = 2^{x}

Prepare

x + 1 = 2^{x}

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Simplify

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

Rewrite

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

3^x+3^{x-2}=270

3^{x} + 3^{x - 2} = 270

20 = 1 0^{x}

3^{x + 5} = 100

12^x=xlog_{10}(12)

1 2^{x} = x lo g_{10} (12)

59 = 2 7^{x}

Frequently Asked Questions (FAQ)

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