Algebra Examples

$\log (\frac{x}{2}) + \log (\frac{20}{x^{2}}) = \log (8)$

Step 1

Simplify the left side.

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Step 1.1

Use the product property of logarithms, $\log_{b} (x) + \log_{b} (y) = \log_{b} (x y)$ .

$\log (\frac{x}{2} \cdot \frac{20}{x^{2}}) = \log (8)$

Step 1.2

Combine.

$\log (\frac{x \cdot 20}{2 x^{2}}) = \log (8)$

Step 1.3

Cancel the common factor of $x$ and $x^{2}$ .

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Step 1.3.1

Factor $x$ out of $x \cdot 20$ .

$\log (\frac{x (20)}{2 x^{2}}) = \log (8)$

Step 1.3.2

Cancel the common factors.

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Step 1.3.2.1

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

$\log (\frac{x (20)}{x (2 x)}) = \log (8)$

Step 1.3.2.2

Cancel the common factor.

$\log (\frac{x \cdot 20}{x (2 x)}) = \log (8)$

Step 1.3.2.3

Rewrite the expression.

$\log (\frac{20}{2 x}) = \log (8)$

Step 1.4

Cancel the common factor of $20$ and $2$ .

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Step 1.4.1

Factor $2$ out of $20$ .

$\log (\frac{2 \cdot 10}{2 x}) = \log (8)$

Step 1.4.2

Cancel the common factors.

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Step 1.4.2.1

Factor $2$ out of $2 x$ .

$\log (\frac{2 \cdot 10}{2 (x)}) = \log (8)$

Step 1.4.2.2

Cancel the common factor.

$\log (\frac{2 \cdot 10}{2 x}) = \log (8)$

Step 1.4.2.3

Rewrite the expression.

$\log (\frac{10}{x}) = \log (8)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{10}{x} = 8$

Step 3

Solve for $x$ .

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Step 3.1

Find the LCD of the terms in the equation.

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Step 3.1.1

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Step 3.1.2

The LCM of one and any expression is the expression.

$x$

Step 3.2

Multiply each term in $\frac{10}{x} = 8$ by $x$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{10}{x} = 8$ by $x$ .

$\frac{10}{x} x = 8 x$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $x$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{10}{x} x = 8 x$

Step 3.2.2.1.2

Rewrite the expression.

$10 = 8 x$

Step 3.3

Solve the equation.

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Step 3.3.1

Rewrite the equation as $8 x = 10$ .

$8 x = 10$

Step 3.3.2

Divide each term in $8 x = 10$ by $8$ and simplify.

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Step 3.3.2.1

Divide each term in $8 x = 10$ by $8$ .

$\frac{8 x}{8} = \frac{10}{8}$

Step 3.3.2.2

Simplify the left side.

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Step 3.3.2.2.1

Cancel the common factor of $8$ .

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Step 3.3.2.2.1.1

Cancel the common factor.

$\frac{8 x}{8} = \frac{10}{8}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{10}{8}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Cancel the common factor of $10$ and $8$ .

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Step 3.3.2.3.1.1

Factor $2$ out of $10$ .

$x = \frac{2 (5)}{8}$

Step 3.3.2.3.1.2

Cancel the common factors.

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Step 3.3.2.3.1.2.1

Factor $2$ out of $8$ .

$x = \frac{2 \cdot 5}{2 \cdot 4}$

Step 3.3.2.3.1.2.2

Cancel the common factor.

$x = \frac{2 \cdot 5}{2 \cdot 4}$

Step 3.3.2.3.1.2.3

Rewrite the expression.

$x = \frac{5}{4}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5}{4}$

Decimal Form:

$x = 1.25$

Mixed Number Form:

$x = 1 \frac{1}{4}$