Algebra Examples

$\log_{36} (x + 4) - \log_{36} (2 x - 14) = \frac{1}{2}$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{36} (\frac{x + 4}{2 x - 14}) = \frac{1}{2}$

Step 1.2

Factor $2$ out of $2 x - 14$ .

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Step 1.2.1

Factor $2$ out of $2 x$ .

$\log_{36} (\frac{x + 4}{2 (x) - 14}) = \frac{1}{2}$

Step 1.2.2

Factor $2$ out of $- 14$ .

$\log_{36} (\frac{x + 4}{2 x + 2 \cdot - 7}) = \frac{1}{2}$

Step 1.2.3

Factor $2$ out of $2 x + 2 \cdot - 7$ .

$\log_{36} (\frac{x + 4}{2 (x - 7)}) = \frac{1}{2}$

Step 2

Rewrite $\log_{36} (\frac{x + 4}{2 (x - 7)}) = \frac{1}{2}$ 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 $b^{y} = x$ .

$36^{\frac{1}{2}} = \frac{x + 4}{2 (x - 7)}$

Step 3

Cross multiply to remove the fraction.

$x + 4 = 36^{\frac{1}{2}} (2 (x - 7))$

Step 4

Simplify $36^{\frac{1}{2}} (2 (x - 7))$ .

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Step 4.1

Simplify the expression.

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Step 4.1.1

Rewrite $36$ as $6^{2}$ .

$x + 4 = {(6^{2})}^{\frac{1}{2}} (2 (x - 7))$

Step 4.1.2

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

$x + 4 = 6^{2 (\frac{1}{2})} (2 (x - 7))$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Cancel the common factor.

$x + 4 = 6^{2 (\frac{1}{2})} (2 (x - 7))$

Step 4.2.2

Rewrite the expression.

$x + 4 = 6^{1} (2 (x - 7))$

Step 4.3

Evaluate the exponent.

$x + 4 = 6 (2 (x - 7))$

Step 4.4

Apply the distributive property.

$x + 4 = 6 (2 x + 2 \cdot - 7)$

Step 4.5

Multiply $2$ by $- 7$ .

$x + 4 = 6 (2 x - 14)$

Step 4.6

Apply the distributive property.

$x + 4 = 6 (2 x) + 6 \cdot - 14$

Step 4.7

Multiply.

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Step 4.7.1

Multiply $2$ by $6$ .

$x + 4 = 12 x + 6 \cdot - 14$

Step 4.7.2

Multiply $6$ by $- 14$ .

$x + 4 = 12 x - 84$

Step 5

Move all terms containing $x$ to the left side of the equation.

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Step 5.1

Subtract $12 x$ from both sides of the equation.

$x + 4 - 12 x = - 84$

Step 5.2

Subtract $12 x$ from $x$ .

$- 11 x + 4 = - 84$

Step 6

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

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Step 6.1

Subtract $4$ from both sides of the equation.

$- 11 x = - 84 - 4$

Step 6.2

Subtract $4$ from $- 84$ .

$- 11 x = - 88$

Step 7

Divide each term in $- 11 x = - 88$ by $- 11$ and simplify.

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Step 7.1

Divide each term in $- 11 x = - 88$ by $- 11$ .

$\frac{- 11 x}{- 11} = \frac{- 88}{- 11}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $- 11$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{- 11 x}{- 11} = \frac{- 88}{- 11}$

Step 7.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 88}{- 11}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 88$ by $- 11$ .

$x = 8$