Algebra Examples

$\log_{2} (6 x) - \log_{2} (\sqrt{x}) = 2$

Step 1

Simplify the left side.

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Step 1.1

Use the quotient property of logarithms, $\log_{b} (x) - \log_{b} (y) = \log_{b} (\frac{x}{y})$ .

$\log_{2} (\frac{6 x}{\sqrt{x}}) = 2$

Step 1.2

Multiply $\frac{6 x}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\log_{2} (\frac{6 x}{\sqrt{x}} \cdot \frac{\sqrt{x}}{\sqrt{x}}) = 2$

Step 1.3

Combine and simplify the denominator.

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Step 1.3.1

Multiply $\frac{6 x}{\sqrt{x}}$ by $\frac{\sqrt{x}}{\sqrt{x}}$ .

$\log_{2} (\frac{6 x \sqrt{x}}{\sqrt{x} \sqrt{x}}) = 2$

Step 1.3.2

Raise $\sqrt{x}$ to the power of $1$ .

$\log_{2} (\frac{6 x \sqrt{x}}{\sqrt{x} \sqrt{x}}) = 2$

Step 1.3.3

Raise $\sqrt{x}$ to the power of $1$ .

$\log_{2} (\frac{6 x \sqrt{x}}{\sqrt{x} \sqrt{x}}) = 2$

Step 1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\log_{2} (\frac{6 x \sqrt{x}}{{\sqrt{x}}^{1 + 1}}) = 2$

Step 1.3.5

Add $1$ and $1$ .

$\log_{2} (\frac{6 x \sqrt{x}}{{\sqrt{x}}^{2}}) = 2$

Step 1.3.6

Rewrite ${\sqrt{x}}^{2}$ as $x$ .

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Step 1.3.6.1

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

$\log_{2} (\frac{6 x \sqrt{x}}{{(x^{\frac{1}{2}})}^{2}}) = 2$

Step 1.3.6.2

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

$\log_{2} (\frac{6 x \sqrt{x}}{x^{\frac{1}{2} \cdot 2}}) = 2$

Step 1.3.6.3

Combine $\frac{1}{2}$ and $2$ .

$\log_{2} (\frac{6 x \sqrt{x}}{x^{\frac{2}{2}}}) = 2$

Step 1.3.6.4

Cancel the common factor of $2$ .

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Step 1.3.6.4.1

Cancel the common factor.

$\log_{2} (\frac{6 x \sqrt{x}}{x^{\frac{2}{2}}}) = 2$

Step 1.3.6.4.2

Rewrite the expression.

$\log_{2} (\frac{6 x \sqrt{x}}{x}) = 2$

Step 1.3.6.5

Simplify.

$\log_{2} (\frac{6 x \sqrt{x}}{x}) = 2$

Step 1.4

Cancel the common factor of $x$ .

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Step 1.4.1

Cancel the common factor.

$\log_{2} (\frac{6 x \sqrt{x}}{x}) = 2$

Step 1.4.2

Divide $6 \sqrt{x}$ by $1$ .

$\log_{2} (6 \sqrt{x}) = 2$

Step 2

Rewrite $\log_{2} (6 \sqrt{x}) = 2$ 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 $b^{y} = x$ .

$2^{2} = 6 \sqrt{x}$

Step 3

Solve for $x$ .

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Step 3.1

Rewrite the equation as $6 \sqrt{x} = 2^{2}$ .

$6 \sqrt{x} = 2^{2}$

Step 3.2

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

${(6 \sqrt{x})}^{2} = {(2^{2})}^{2}$

Step 3.3

Simplify each side of the equation.

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Step 3.3.1

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

${(6 x^{\frac{1}{2}})}^{2} = {(2^{2})}^{2}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Simplify ${(6 x^{\frac{1}{2}})}^{2}$ .

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Step 3.3.2.1.1

Apply the product rule to $6 x^{\frac{1}{2}}$ .

$6^{2} {(x^{\frac{1}{2}})}^{2} = {(2^{2})}^{2}$

Step 3.3.2.1.2

Raise $6$ to the power of $2$ .

$36 {(x^{\frac{1}{2}})}^{2} = {(2^{2})}^{2}$

Step 3.3.2.1.3

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

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Step 3.3.2.1.3.1

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

$36 x^{\frac{1}{2} \cdot 2} = {(2^{2})}^{2}$

Step 3.3.2.1.3.2

Cancel the common factor of $2$ .

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Step 3.3.2.1.3.2.1

Cancel the common factor.

$36 x^{\frac{1}{2} \cdot 2} = {(2^{2})}^{2}$

Step 3.3.2.1.3.2.2

Rewrite the expression.

$36 x^{1} = {(2^{2})}^{2}$

Step 3.3.2.1.4

Simplify.

$36 x = {(2^{2})}^{2}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

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

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Step 3.3.3.1.1

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

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Step 3.3.3.1.1.1

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

$36 x = 2^{2 \cdot 2}$

Step 3.3.3.1.1.2

Multiply $2$ by $2$ .

$36 x = 2^{4}$

Step 3.3.3.1.2

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

$36 x = 16$

Step 3.4

Divide each term in $36 x = 16$ by $36$ and simplify.

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Step 3.4.1

Divide each term in $36 x = 16$ by $36$ .

$\frac{36 x}{36} = \frac{16}{36}$

Step 3.4.2

Simplify the left side.

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Step 3.4.2.1

Cancel the common factor of $36$ .

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Step 3.4.2.1.1

Cancel the common factor.

$\frac{36 x}{36} = \frac{16}{36}$

Step 3.4.2.1.2

Divide $x$ by $1$ .

$x = \frac{16}{36}$

Step 3.4.3

Simplify the right side.

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Step 3.4.3.1

Cancel the common factor of $16$ and $36$ .

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Step 3.4.3.1.1

Factor $4$ out of $16$ .

$x = \frac{4 (4)}{36}$

Step 3.4.3.1.2

Cancel the common factors.

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Step 3.4.3.1.2.1

Factor $4$ out of $36$ .

$x = \frac{4 \cdot 4}{4 \cdot 9}$

Step 3.4.3.1.2.2

Cancel the common factor.

$x = \frac{4 \cdot 4}{4 \cdot 9}$

Step 3.4.3.1.2.3

Rewrite the expression.

$x = \frac{4}{9}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{4}{9}$

Decimal Form:

$x = 0.\overline{4}$