Algebra Examples

$\log_{4} (2 x + 4) - 3 = \log_{4} (3)$

Step 1

Move all the terms containing a logarithm to the left side of the equation.

$\log_{4} (2 x + 4) - \log_{4} (3) = 3$

Step 2

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

$\log_{4} (\frac{2 x + 4}{3}) = 3$

Step 3

Factor $2$ out of $2 x + 4$ .

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Step 3.1

Factor $2$ out of $2 x$ .

$\log_{4} (\frac{2 (x) + 4}{3}) = 3$

Step 3.2

Factor $2$ out of $4$ .

$\log_{4} (\frac{2 x + 2 \cdot 2}{3}) = 3$

Step 3.3

Factor $2$ out of $2 x + 2 \cdot 2$ .

$\log_{4} (\frac{2 (x + 2)}{3}) = 3$

Step 4

Rewrite $\log_{4} (\frac{2 (x + 2)}{3}) = 3$ 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 $b^{y} = x$ .

$4^{3} = \frac{2 (x + 2)}{3}$

Step 5

Solve for $x$ .

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Step 5.1

Rewrite the equation as $\frac{2 (x + 2)}{3} = 4^{3}$ .

$\frac{2 (x + 2)}{3} = 4^{3}$

Step 5.2

Multiply both sides of the equation by $\frac{3}{2}$ .

$\frac{3}{2} \cdot \frac{2 (x + 2)}{3} = \frac{3}{2} \cdot 4^{3}$

Step 5.3

Simplify both sides of the equation.

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Step 5.3.1

Simplify the left side.

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Step 5.3.1.1

Simplify $\frac{3}{2} \cdot \frac{2 (x + 2)}{3}$ .

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Step 5.3.1.1.1

Cancel the common factor of $3$ .

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Step 5.3.1.1.1.1

Cancel the common factor.

$\frac{3}{2} \cdot \frac{2 (x + 2)}{3} = \frac{3}{2} \cdot 4^{3}$

Step 5.3.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 (x + 2)) = \frac{3}{2} \cdot 4^{3}$

Step 5.3.1.1.2

Cancel the common factor of $2$ .

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Step 5.3.1.1.2.1

Cancel the common factor.

$\frac{1}{2} (2 (x + 2)) = \frac{3}{2} \cdot 4^{3}$

Step 5.3.1.1.2.2

Rewrite the expression.

$x + 2 = \frac{3}{2} \cdot 4^{3}$

Step 5.3.2

Simplify the right side.

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Step 5.3.2.1

Simplify $\frac{3}{2} \cdot 4^{3}$ .

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Step 5.3.2.1.1

Raise $4$ to the power of $3$ .

$x + 2 = \frac{3}{2} \cdot 64$

Step 5.3.2.1.2

Cancel the common factor of $2$ .

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Step 5.3.2.1.2.1

Factor $2$ out of $64$ .

$x + 2 = \frac{3}{2} \cdot (2 (32))$

Step 5.3.2.1.2.2

Cancel the common factor.

$x + 2 = \frac{3}{2} \cdot (2 \cdot 32)$

Step 5.3.2.1.2.3

Rewrite the expression.

$x + 2 = 3 \cdot 32$

Step 5.3.2.1.3

Multiply $3$ by $32$ .

$x + 2 = 96$

Step 5.4

Move all terms not containing $x$ to the right side of the equation.

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Step 5.4.1

Subtract $2$ from both sides of the equation.

$x = 96 - 2$

Step 5.4.2

Subtract $2$ from $96$ .

$x = 94$