Precalculus Examples

$\log_{225} (5 x + 5) - \log_{225} (5 x - 9) = \frac{1}{2}$

Step 1

Simplify the left side.

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Step 1.1

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

$\log_{225} (\frac{5 x + 5}{5 x - 9}) = \frac{1}{2}$

Step 1.2

Factor $5$ out of $5 x + 5$ .

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Step 1.2.1

Factor $5$ out of $5 x$ .

$\log_{225} (\frac{5 (x) + 5}{5 x - 9}) = \frac{1}{2}$

Step 1.2.2

Factor $5$ out of $5$ .

$\log_{225} (\frac{5 (x) + 5 (1)}{5 x - 9}) = \frac{1}{2}$

Step 1.2.3

Factor $5$ out of $5 (x) + 5 (1)$ .

$\log_{225} (\frac{5 (x + 1)}{5 x - 9}) = \frac{1}{2}$

Step 2

Rewrite $\log_{225} (\frac{5 (x + 1)}{5 x - 9}) = \frac{1}{2}$ 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 $b^{y} = x$ .

$225^{\frac{1}{2}} = \frac{5 (x + 1)}{5 x - 9}$

Step 3

Cross multiply to remove the fraction.

$5 (x + 1) = 225^{\frac{1}{2}} (5 x - 9)$

Step 4

Simplify $225^{\frac{1}{2}} (5 x - 9)$ .

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Step 4.1

Simplify the expression.

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Step 4.1.1

Rewrite $225$ as $15^{2}$ .

$5 (x + 1) = {(15^{2})}^{\frac{1}{2}} (5 x - 9)$

Step 4.1.2

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

$5 (x + 1) = 15^{2 (\frac{1}{2})} (5 x - 9)$

Step 4.2

Cancel the common factor of $2$ .

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Step 4.2.1

Cancel the common factor.

$5 (x + 1) = 15^{2 (\frac{1}{2})} (5 x - 9)$

Step 4.2.2

Rewrite the expression.

$5 (x + 1) = 15^{1} (5 x - 9)$

Step 4.3

Evaluate the exponent.

$5 (x + 1) = 15 (5 x - 9)$

Step 4.4

Apply the distributive property.

$5 (x + 1) = 15 (5 x) + 15 \cdot - 9$

Step 4.5

Multiply.

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Step 4.5.1

Multiply $5$ by $15$ .

$5 (x + 1) = 75 x + 15 \cdot - 9$

Step 4.5.2

Multiply $15$ by $- 9$ .

$5 (x + 1) = 75 x - 135$

Step 5

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

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Step 5.1

Subtract $75 x$ from both sides of the equation.

$5 (x + 1) - 75 x = - 135$

Step 5.2

Simplify each term.

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Step 5.2.1

Apply the distributive property.

$5 x + 5 \cdot 1 - 75 x = - 135$

Step 5.2.2

Multiply $5$ by $1$ .

$5 x + 5 - 75 x = - 135$

Step 5.3

Subtract $75 x$ from $5 x$ .

$- 70 x + 5 = - 135$

Step 6

Factor $5$ out of $- 70 x + 5$ .

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Step 6.1

Factor $5$ out of $- 70 x$ .

$5 (- 14 x) + 5 = - 135$

Step 6.2

Factor $5$ out of $5$ .

$5 (- 14 x) + 5 (1) = - 135$

Step 6.3

Factor $5$ out of $5 (- 14 x) + 5 (1)$ .

$5 (- 14 x + 1) = - 135$

Step 7

Divide each term in $5 (- 14 x + 1) = - 135$ by $5$ and simplify.

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Step 7.1

Divide each term in $5 (- 14 x + 1) = - 135$ by $5$ .

$\frac{5 (- 14 x + 1)}{5} = \frac{- 135}{5}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Cancel the common factor of $5$ .

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Step 7.2.1.1

Cancel the common factor.

$\frac{5 (- 14 x + 1)}{5} = \frac{- 135}{5}$

Step 7.2.1.2

Divide $- 14 x + 1$ by $1$ .

$- 14 x + 1 = \frac{- 135}{5}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Divide $- 135$ by $5$ .

$- 14 x + 1 = - 27$

Step 8

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

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Step 8.1

Subtract $1$ from both sides of the equation.

$- 14 x = - 27 - 1$

Step 8.2

Subtract $1$ from $- 27$ .

$- 14 x = - 28$

Step 9

Divide each term in $- 14 x = - 28$ by $- 14$ and simplify.

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Step 9.1

Divide each term in $- 14 x = - 28$ by $- 14$ .

$\frac{- 14 x}{- 14} = \frac{- 28}{- 14}$

Step 9.2

Simplify the left side.

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Step 9.2.1

Cancel the common factor of $- 14$ .

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Step 9.2.1.1

Cancel the common factor.

$\frac{- 14 x}{- 14} = \frac{- 28}{- 14}$

Step 9.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 28}{- 14}$

Step 9.3

Simplify the right side.

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Step 9.3.1

Divide $- 28$ by $- 14$ .

$x = 2$