Solve for x 2 log of (2x)^4=16

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

Step 1

Divide each term in $2 \log ({(2 x)}^{4}) = 16$ by $2$ and simplify.

Divide each term in $2 \log ({(2 x)}^{4}) = 16$ by $2$ .

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $\log ({(2 x)}^{4})$ by $1$ .

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

Simplify the expression.

Apply the product rule to $2 x$ .

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

Raise $2$ to the power of $4$ .

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

Simplify the right side.

Divide $16$ by $2$ .

$\log (16 x^{4}) = 8$

Step 2

Rewrite $\log (16 x^{4}) = 8$ 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 $b^{y} = x$ .

$10^{8} = 16 x^{4}$

Step 3

Solve for $x$ .

Rewrite the equation as $16 x^{4} = 10^{8}$ .

$16 x^{4} = 10^{8}$

Divide each term in $16 x^{4} = 10^{8}$ by $16$ and simplify.

Divide each term in $16 x^{4} = 10^{8}$ by $16$ .

$\frac{16 x^{4}}{16} = \frac{10^{8}}{16}$

Simplify the left side.

Cancel the common factor of $16$ .

Cancel the common factor.

$\frac{16 x^{4}}{16} = \frac{10^{8}}{16}$

Divide $x^{4}$ by $1$ .

$x^{4} = \frac{10^{8}}{16}$

Simplify the right side.

Raise $10$ to the power of $8$ .

$x^{4} = \frac{100000000}{16}$

Divide $100000000$ by $16$ .

$x^{4} = 6250000$

Take the 4th root of both sides of the equation to eliminate the exponent on the left side.

$x = \pm \sqrt[4]{6250000}$

Simplify $\pm \sqrt[4]{6250000}$ .

Rewrite $6250000$ as $50^{4}$ .

$x = \pm \sqrt[4]{50^{4}}$

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

$x = \pm 50$

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.

$x = 50$

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

$x = - 50$

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

$x = 50, - 50$

Solve for x 2 log of (2x)^4=16

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