What is the answer to (k+1)2^k=32 ?

The answer to (k+1)2^k=32 is k=3

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(k+1)2^k=32

(k + 1) 2^{k} = 32

k = 3

Solution steps

(k + 1) \cdot 2^{k} = 32

Prepare

(k + 1) 2^{k} = 32

for Lambert form:

(k + 1) e^{l n (2) k} = 32

Rewrite the equation with

(k + 1) ln (2) = u

and

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

((\frac{u}{ln ( 2 )} - 1) + 1) e^{l n (2) (\frac{u}{l n ( 2 )} - 1)} = 32

Simplify

\frac{e ^{u - l n (2)} u}{ln ( 2 )} = 32

Rewrite

\frac{e ^{u - l n (2)} u}{ln ( 2 )} = 32

in Lambert form:

e^{u} u = 64 ln (2)

Solve

e^{u} u = 64 ln (2) : u = 4 ln (2)

u = 4 ln (2)

Substitute back

u = (k + 1) ln (2),

solve for

k

Solve

(k + 1) ln (2) = 4 ln (2) : k = 3

k = 3

\frac{1}{2 ^{x - 4}} - \frac{1}{2 ^{x - 2}} = 3

2^{x} = \frac{1}{2}

3^{x} = 5^{x - 2}

3^{x + 2} - 3^{x} = 72

8^{x} - 4^{x} - 2^{x} + 1 + 2 = 0