What is the answer to 2^{x-2}+1=x ?

The answer to 2^{x-2}+1=x is x=3,x=2

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2^{x-2}+1=x

2^{x - 2} + 1 = x

x = 3, x = 2

Solution steps

2^{x - 2} + 1 = x

Prepare

2^{x - 2} + 1 = x

for Lambert form:

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

Rewrite the equation with

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

and

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

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

Simplify

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

Rewrite

1 = e^{u + l n (2)} (- \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 = 3

Solve

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

x = 3, x = 2

3 e^{0.007 x} = 2.4 e^{0.06 x}

e^{4 x} e^{- 2 x + 1} = 2

5^{2 x - 1} = 0

- 7 \cdot 2 0^{x - 4} = - 49

1240000 = 525600 e^{0.03 t}