Calculus Examples

$\log (10 x) - \log (2 x - 5) = \log (9)$

Step 1

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

$\log (\frac{10 x}{2 x - 5}) = \log (9)$

Step 2

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

$\frac{10 x}{2 x - 5} = 9$

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.

$2 x - 5, 1$

Step 3.1.2

Remove parentheses.

$2 x - 5, 1$

Step 3.1.3

The LCM of one and any expression is the expression.

$2 x - 5$

Step 3.2

Multiply each term in $\frac{10 x}{2 x - 5} = 9$ by $2 x - 5$ to eliminate the fractions.

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Step 3.2.1

Multiply each term in $\frac{10 x}{2 x - 5} = 9$ by $2 x - 5$ .

$\frac{10 x}{2 x - 5} (2 x - 5) = 9 (2 x - 5)$

Step 3.2.2

Simplify the left side.

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Step 3.2.2.1

Cancel the common factor of $2 x - 5$ .

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Step 3.2.2.1.1

Cancel the common factor.

$\frac{10 x}{2 x - 5} (2 x - 5) = 9 (2 x - 5)$

Step 3.2.2.1.2

Rewrite the expression.

$10 x = 9 (2 x - 5)$

Step 3.2.3

Simplify the right side.

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Step 3.2.3.1

Apply the distributive property.

$10 x = 9 (2 x) + 9 \cdot - 5$

Step 3.2.3.2

Multiply.

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Step 3.2.3.2.1

Multiply $2$ by $9$ .

$10 x = 18 x + 9 \cdot - 5$

Step 3.2.3.2.2

Multiply $9$ by $- 5$ .

$10 x = 18 x - 45$

Step 3.3

Solve the equation.

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Step 3.3.1

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

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Step 3.3.1.1

Subtract $18 x$ from both sides of the equation.

$10 x - 18 x = - 45$

Step 3.3.1.2

Subtract $18 x$ from $10 x$ .

$- 8 x = - 45$

Step 3.3.2

Divide each term in $- 8 x = - 45$ by $- 8$ and simplify.

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Step 3.3.2.1

Divide each term in $- 8 x = - 45$ by $- 8$ .

$\frac{- 8 x}{- 8} = \frac{- 45}{- 8}$

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.

$\frac{- 8 x}{- 8} = \frac{- 45}{- 8}$

Step 3.3.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 45}{- 8}$

Step 3.3.2.3

Simplify the right side.

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Step 3.3.2.3.1

Dividing two negative values results in a positive value.

$x = \frac{45}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{45}{8}$

Decimal Form:

$x = 5.625$

Mixed Number Form:

$x = 5 \frac{5}{8}$