What is the answer to 12x*3^{x+1}=648 ?

The answer to 12x*3^{x+1}=648 is x=2

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12x*3^{x+1}=648

12 x \cdot 3^{x + 1} = 648

x = 2

Solution steps

12 x \cdot 3^{x + 1} = 648

Prepare

12 x \cdot 3^{x + 1} = 648

for Lambert form:

12 x e^{l n (3) (x + 1)} = 648

Rewrite the equation with

x ln (3) = u

and

x = \frac{u}{ln ( 3 )}

12 (\frac{u}{ln ( 3 )}) e^{l n (3) ((\frac{u}{l n ( 3 )}) + 1)} = 648

Simplify

\frac{12 e ^{u + l n (3)} u}{ln ( 3 )} = 648

Rewrite

\frac{12 e ^{u + l n (3)} u}{ln ( 3 )} = 648

in Lambert form:

e^{u} u = 18 ln (3)

Solve

e^{u} u = 18 ln (3) : u = 2 ln (3)

u = 2 ln (3)

Substitute back

u = x ln (3),

solve for

x

Solve

x ln (3) = 2 ln (3) : x = 2

x = 2

2 7^{x + 5} = 81

6 \cdot 2^{x} = 14

5 + e^{x} = - \frac{x}{2} + 2

5^{3 x} = 2 5^{x + 1}

1 6^{2 x} = \frac{1}{8}