Algebra Examples

$\log_{2} (\log_{2} (\sqrt{4 x})) = 1$

Step 1

Rewrite $\log_{2} (\log_{2} (\sqrt{4 x})) = 1$ 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 $b^{y} = x$ .

$2^{1} = \log_{2} (\sqrt{4 x})$

Step 2

Solve for $x$ .

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Step 2.1

Rewrite the equation as $\log_{2} (\sqrt{4 x}) = 2^{1}$ .

$\log_{2} (\sqrt{4 x}) = 2$

Step 2.2

Rewrite $\log_{2} (\sqrt{4 x}) = 2^{1}$ 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 $b^{y} = x$ .

$2^{2^{1}} = \sqrt{4 x}$

Step 2.3

Solve for $x$ .

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Step 2.3.1

Rewrite the equation as $\sqrt{4 x} = 2^{2^{1}}$ .

$\sqrt{4 x} = 2^{2^{1}}$

Step 2.3.2

To remove the radical on the left side of the equation, square both sides of the equation.

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

Step 2.3.3

Simplify each side of the equation.

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Step 2.3.3.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{4 x}$ as ${(4 x)}^{\frac{1}{2}}$ .

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

Step 2.3.3.2

Simplify the left side.

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Step 2.3.3.2.1

Simplify ${({(4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.3.2.1.1

Multiply the exponents in ${({(4 x)}^{\frac{1}{2}})}^{2}$ .

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Step 2.3.3.2.1.1.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 2.3.3.2.1.1.2

Cancel the common factor of $2$ .

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Step 2.3.3.2.1.1.2.1

Cancel the common factor.

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

Step 2.3.3.2.1.1.2.2

Rewrite the expression.

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

Step 2.3.3.2.1.2

Simplify.

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

Step 2.3.3.3

Simplify the right side.

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Step 2.3.3.3.1

Simplify ${(2^{2^{1}})}^{2}$ .

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Step 2.3.3.3.1.1

Evaluate the exponent.

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

Step 2.3.3.3.1.2

Multiply the exponents in ${(2^{2})}^{2}$ .

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Step 2.3.3.3.1.2.1

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$4 x = 2^{2 \cdot 2}$

Step 2.3.3.3.1.2.2

Multiply $2$ by $2$ .

$4 x = 2^{4}$

Step 2.3.3.3.1.3

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

$4 x = 16$

Step 2.3.4

Divide each term in $4 x = 16$ by $4$ and simplify.

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Step 2.3.4.1

Divide each term in $4 x = 16$ by $4$ .

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

Step 2.3.4.2

Simplify the left side.

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Step 2.3.4.2.1

Cancel the common factor of $4$ .

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Step 2.3.4.2.1.1

Cancel the common factor.

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

Step 2.3.4.2.1.2

Divide $x$ by $1$ .

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

Step 2.3.4.3

Simplify the right side.

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Step 2.3.4.3.1

Divide $16$ by $4$ .

$x = 4$