Solve for x log base 2x of 16=-2

$\log_{2 x} (16) = - 2$

Rewrite $\log_{2 x} (16) = - 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b \neq 1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

${(2 x)}^{- 2} = 16$

Solve for $x$ .

Simplify ${(2 x)}^{- 2}$ .

Rewrite the expression using the negative exponent rule $b^{- n} = \frac{1}{b^{n}}$ .

$\frac{1}{{(2 x)}^{2}} = 16$

Simplify the denominator.

Apply the product rule to $2 x$ .

$\frac{1}{2^{2} x^{2}} = 16$

Raise $2$ to the power of $2$ .

$\frac{1}{4 x^{2}} = 16$

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$4 x^{2}, 1$

The LCM of one and any expression is the expression.

$4 x^{2}$

Multiply each term in $\frac{1}{4 x^{2}} = 16$ by $4 x^{2}$ to eliminate the fractions.

Multiply each term in $\frac{1}{4 x^{2}} = 16$ by $4 x^{2}$ .

$\frac{1}{4 x^{2}} (4 x^{2}) = 16 (4 x^{2})$

Simplify the left side.

Rewrite using the commutative property of multiplication.

$4 \frac{1}{4 x^{2}} x^{2} = 16 (4 x^{2})$

Cancel the common factor of $4$ .

Factor $4$ out of $4 x^{2}$ .

$4 \frac{1}{4 (x^{2})} x^{2} = 16 (4 x^{2})$

Cancel the common factor.

$4 \frac{1}{4 x^{2}} x^{2} = 16 (4 x^{2})$

Rewrite the expression.

$\frac{1}{x^{2}} x^{2} = 16 (4 x^{2})$

Cancel the common factor of $x^{2}$ .

Cancel the common factor.

$\frac{1}{x^{2}} x^{2} = 16 (4 x^{2})$

Rewrite the expression.

$1 = 16 (4 x^{2})$

Simplify the right side.

Multiply $4$ by $16$ .

$1 = 64 x^{2}$

Solve the equation.

Rewrite the equation as $64 x^{2} = 1$ .

$64 x^{2} = 1$

Divide each term in $64 x^{2} = 1$ by $64$ and simplify.

Divide each term in $64 x^{2} = 1$ by $64$ .

$\frac{64 x^{2}}{64} = \frac{1}{64}$

Simplify the left side.

Cancel the common factor of $64$ .

Cancel the common factor.

$\frac{64 x^{2}}{64} = \frac{1}{64}$

Divide $x^{2}$ by $1$ .

$x^{2} = \frac{1}{64}$

Take the square root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt{\frac{1}{64}}$

Simplify $\pm \sqrt{\frac{1}{64}}$ .

Rewrite $\sqrt{\frac{1}{64}}$ as $\frac{\sqrt{1}}{\sqrt{64}}$ .

$x = \pm \frac{\sqrt{1}}{\sqrt{64}}$

Any root of $1$ is $1$ .

$x = \pm \frac{1}{\sqrt{64}}$

Simplify the denominator.

Rewrite $64$ as $8^{2}$ .

$x = \pm \frac{1}{\sqrt{8^{2}}}$

Pull terms out from under the radical, assuming positive real numbers.

$x = \pm \frac{1}{8}$

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$x = \frac{1}{8}$

Next, use the negative value of the $\pm$ to find the second solution.

$x = - \frac{1}{8}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{1}{8}, - \frac{1}{8}$

Exclude the solutions that do not make $\log_{2 x} (16) = - 2$ true.

$x = \frac{1}{8}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{8}$

Decimal Form:

$x = 0.125$

Solve for x log base 2x of 16=-2

Related Questions

Support LCMGCF.com by disabling your adblocker.