What is the answer to 8x=16^x ?

The answer to 8x=16^x is x= 1/2 ,x= 1/4

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8x=16^x

8 x = 1 6^{x}

x = \frac{1}{2}, x = \frac{1}{4}

Decimal Notation

x = 0.5, x = 0.25

Solution steps

8 x = 1 6^{x}

Prepare

8 x = 1 6^{x}

for Lambert form:

e^{l n (2) (3 - 4 x)} x = 1

Rewrite the equation with

- 4 x ln (2) = u

and

x = - \frac{u}{4 ln ( 2 )}

e^{l n (2) (3 - 4 (- \frac{u}{4 l n ( 2 )}))} (- \frac{u}{4 ln ( 2 )}) = 1

Simplify

- \frac{e ^{3 l n (2) + u} u}{4 ln ( 2 )} = 1

Rewrite

- \frac{e ^{3 l n (2) + u} u}{4 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 = - 4 x ln (2),

solve for

x

Solve

- 4 x ln (2) = - 2 ln (2) : x = \frac{1}{2}

Solve

- 4 x ln (2) = - ln (2) : x = \frac{1}{4}

x = \frac{1}{2}, x = \frac{1}{4}

0.9 = \frac{0.1}{0.5 5 ^{x}}

(3^{x}) (4^{2 x + 1}) = 6^{x + 2}

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

5^{2 x} - 5^{x} - 12 = 0

e^{x} + e^{x} x = 0