Algebra Examples

$\ln (x^{2} - 5 x) - \ln (x^{2} - 25) = 3$

Step 1

Simplify the left side.

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Step 1.1

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

$\ln (\frac{x^{2} - 5 x}{x^{2} - 25}) = 3$

Step 1.2

Factor $x$ out of $x^{2} - 5 x$ .

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Step 1.2.1

Factor $x$ out of $x^{2}$ .

$\ln (\frac{x \cdot x - 5 x}{x^{2} - 25}) = 3$

Step 1.2.2

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

$\ln (\frac{x \cdot x + x \cdot - 5}{x^{2} - 25}) = 3$

Step 1.2.3

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$\ln (\frac{x (x - 5)}{x^{2} - 25}) = 3$

Step 1.3

Simplify the denominator.

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Step 1.3.1

Rewrite $25$ as $5^{2}$ .

$\ln (\frac{x (x - 5)}{x^{2} - 5^{2}}) = 3$

Step 1.3.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = x$ and $b = 5$ .

$\ln (\frac{x (x - 5)}{(x + 5) (x - 5)}) = 3$

Step 1.4

Cancel the common factor of $x - 5$ .

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Step 1.4.1

Cancel the common factor.

$\ln (\frac{x (x - 5)}{(x + 5) (x - 5)}) = 3$

Step 1.4.2

Rewrite the expression.

$\ln (\frac{x}{x + 5}) = 3$

Step 2

Rewrite $\ln (\frac{x}{x + 5}) = 3$ 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 $b^{y} = x$ .

$e^{3} = \frac{x}{x + 5}$

Step 3

Cross multiply to remove the fraction.

$x = e^{3} (x + 5)$

Step 4

Simplify $e^{3} (x + 5)$ .

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Step 4.1

Apply the distributive property.

$x = e^{3} x + e^{3} \cdot 5$

Step 4.2

Move $5$ to the left of $e^{3}$ .

$x = e^{3} x + 5 e^{3}$

Step 5

Subtract $e^{3} x$ from both sides of the equation.

$x - e^{3} x = 5 e^{3}$

Step 6

Factor the left side of the equation.

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Step 6.1

Factor $x$ out of $x - e^{3} x$ .

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Step 6.1.1

Raise $x$ to the power of $1$ .

$x - e^{3} x = 5 e^{3}$

Step 6.1.2

Factor $x$ out of $x^{1}$ .

$x \cdot 1 - e^{3} x = 5 e^{3}$

Step 6.1.3

Factor $x$ out of $- e^{3} x$ .

$x \cdot 1 + x (- e^{3}) = 5 e^{3}$

Step 6.1.4

Factor $x$ out of $x \cdot 1 + x (- e^{3})$ .

$x (1 - e^{3}) = 5 e^{3}$

Step 6.2

Rewrite $1$ as $1^{3}$ .

$x (1^{3} - e^{3}) = 5 e^{3}$

Step 6.3

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = e$ .

$x ((1 - e) (1^{2} + 1 e + e^{2})) = 5 e^{3}$

Step 6.4

Factor.

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Step 6.4.1

Simplify.

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Step 6.4.1.1

One to any power is one.

$x ((1 - e) (1 + 1 e + e^{2})) = 5 e^{3}$

Step 6.4.1.2

Multiply $e$ by $1$ .

$x ((1 - e) (1 + e + e^{2})) = 5 e^{3}$

Step 6.4.2

Remove unnecessary parentheses.

$x (1 - e) (1 + e + e^{2}) = 5 e^{3}$

Step 7

Divide each term in $x (1 - e) (1 + e + e^{2}) = 5 e^{3}$ by $1 - e^{3}$ and simplify.

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Step 7.1

Divide each term in $x (1 - e) (1 + e + e^{2}) = 5 e^{3}$ by $1 - e^{3}$ .

$\frac{x (1 - e) (1 + e + e^{2})}{1 - e^{3}} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2

Simplify the left side.

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Step 7.2.1

Simplify the denominator.

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Step 7.2.1.1

Rewrite $1$ as $1^{3}$ .

$\frac{x (1 - e) (1 + e + e^{2})}{1^{3} - e^{3}} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = e$ .

$\frac{x (1 - e) (1 + e + e^{2})}{(1 - e) (1^{2} + 1 e + e^{2})} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2.1.3

Simplify.

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Step 7.2.1.3.1

One to any power is one.

$\frac{x (1 - e) (1 + e + e^{2})}{(1 - e) (1 + 1 e + e^{2})} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2.1.3.2

Multiply $e$ by $1$ .

$\frac{x (1 - e) (1 + e + e^{2})}{(1 - e) (1 + e + e^{2})} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2.2

Reduce the expression by cancelling the common factors.

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Step 7.2.2.1

Cancel the common factor of $1 - e$ .

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Step 7.2.2.1.1

Cancel the common factor.

$\frac{x (1 - e) (1 + e + e^{2})}{(1 - e) (1 + e + e^{2})} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2.2.1.2

Rewrite the expression.

$\frac{x (1 + e + e^{2})}{1 + e + e^{2}} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2.2.2

Cancel the common factor of $1 + e + e^{2}$ .

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Step 7.2.2.2.1

Cancel the common factor.

$\frac{x (1 + e + e^{2})}{1 + e + e^{2}} = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.2.2.2.2

Divide $x$ by $1$ .

$x = \frac{5 e^{3}}{1 - e^{3}}$

Step 7.3

Simplify the right side.

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Step 7.3.1

Simplify the denominator.

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Step 7.3.1.1

Rewrite $1$ as $1^{3}$ .

$x = \frac{5 e^{3}}{1^{3} - e^{3}}$

Step 7.3.1.2

Since both terms are perfect cubes, factor using the difference of cubes formula, $a^{3} - b^{3} = (a - b) (a^{2} + a b + b^{2})$ where $a = 1$ and $b = e$ .

$x = \frac{5 e^{3}}{(1 - e) (1^{2} + 1 e + e^{2})}$

Step 7.3.1.3

Simplify.

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Step 7.3.1.3.1

One to any power is one.

$x = \frac{5 e^{3}}{(1 - e) (1 + 1 e + e^{2})}$

Step 7.3.1.3.2

Multiply $e$ by $1$ .

$x = \frac{5 e^{3}}{(1 - e) (1 + e + e^{2})}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 e^{3}}{(1 - e) (1 + e + e^{2})}$

Decimal Form:

$x = - 5.26197848 \dots$