Finite Math Examples

$\log (6) + \log (x - 6) = \log (x)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (6 (x - 6)) = \log (x)$

Step 1.2

Apply the distributive property.

$\log (6 x + 6 \cdot - 6) = \log (x)$

Step 1.3

Multiply $6$ by $- 6$ .

$\log (6 x - 36) = \log (x)$

Step 2

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

$6 x - 36 = x$

Step 3

Solve for $x$ .

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Step 3.1

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

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Step 3.1.1

Subtract $x$ from both sides of the equation.

$6 x - 36 - x = 0$

Step 3.1.2

Subtract $x$ from $6 x$ .

$5 x - 36 = 0$

Step 3.2

Add $36$ to both sides of the equation.

$5 x = 36$

Step 3.3

Divide each term in $5 x = 36$ by $5$ and simplify.

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Step 3.3.1

Divide each term in $5 x = 36$ by $5$ .

$\frac{5 x}{5} = \frac{36}{5}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Cancel the common factor of $5$ .

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Step 3.3.2.1.1

Cancel the common factor.

$\frac{5 x}{5} = \frac{36}{5}$

Step 3.3.2.1.2

Divide $x$ by $1$ .

$x = \frac{36}{5}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{36}{5}$

Decimal Form:

$x = 7.2$

Mixed Number Form:

$x = 7 \frac{1}{5}$