Algebra Examples

$\log (2 x) - \log (5) = \log (30)$

Step 1

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

$\log (\frac{2 x}{5}) = \log (30)$

Step 2

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

$\frac{2 x}{5} = 30$

Step 3

Solve for $x$ .

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Step 3.1

Multiply both sides of the equation by $\frac{5}{2}$ .

$\frac{5}{2} \cdot \frac{2 x}{5} = \frac{5}{2} \cdot 30$

Step 3.2

Simplify both sides of the equation.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{5}{2} \cdot \frac{2 x}{5}$ .

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Step 3.2.1.1.1

Cancel the common factor of $5$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{5}{2} \cdot \frac{2 x}{5} = \frac{5}{2} \cdot 30$

Step 3.2.1.1.1.2

Rewrite the expression.

$\frac{1}{2} (2 x) = \frac{5}{2} \cdot 30$

Step 3.2.1.1.2

Cancel the common factor of $2$ .

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Step 3.2.1.1.2.1

Factor $2$ out of $2 x$ .

$\frac{1}{2} (2 (x)) = \frac{5}{2} \cdot 30$

Step 3.2.1.1.2.2

Cancel the common factor.

$\frac{1}{2} (2 x) = \frac{5}{2} \cdot 30$

Step 3.2.1.1.2.3

Rewrite the expression.

$x = \frac{5}{2} \cdot 30$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Simplify $\frac{5}{2} \cdot 30$ .

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Step 3.2.2.1.1

Cancel the common factor of $2$ .

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Step 3.2.2.1.1.1

Factor $2$ out of $30$ .

$x = \frac{5}{2} \cdot (2 (15))$

Step 3.2.2.1.1.2

Cancel the common factor.

$x = \frac{5}{2} \cdot (2 \cdot 15)$

Step 3.2.2.1.1.3

Rewrite the expression.

$x = 5 \cdot 15$

Step 3.2.2.1.2

Multiply $5$ by $15$ .

$x = 75$