Precalculus Examples

$\log (9) (x - 2) + \log (9) (x - 2) = 1$

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\log (9) x + \log (9) \cdot - 2 + \log (9) (x - 2) = 1$

Step 1.1.2

Move $- 2$ to the left of $\log (9)$ .

$\log (9) x - 2 \cdot \log (9) + \log (9) (x - 2) = 1$

Step 1.1.3

Apply the distributive property.

$\log (9) x - 2 \log (9) + \log (9) x + \log (9) \cdot - 2 = 1$

Step 1.1.4

Move $- 2$ to the left of $\log (9)$ .

$\log (9) x - 2 \log (9) + \log (9) x - 2 \log (9) = 1$

Step 1.2

Simplify by adding terms.

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Step 1.2.1

Add $\log (9) x$ and $\log (9) x$ .

$2 \log (9) x - 2 \log (9) - 2 \log (9) = 1$

Step 1.2.2

Subtract $2 \log (9)$ from $- 2 \log (9)$ .

$2 \log (9) x - 4 \log (9) = 1$

Step 1.2.3

Reorder factors in $2 \log (9) x - 4 \log (9)$ .

$2 x \log (9) - 4 \log (9) = 1$

Step 2

Simplify the left side.

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Step 2.1

Simplify each term.

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Step 2.1.1

Simplify $2 \log (9)$ by moving $2$ inside the logarithm.

$x \log (9^{2}) - 4 \log (9) = 1$

Step 2.1.2

Raise $9$ to the power of $2$ .

$x \log (81) - 4 \log (9) = 1$

Step 2.1.3

Simplify $- 4 \log (9)$ by moving $4$ inside the logarithm.

$x \log (81) - \log (9^{4}) = 1$

Step 2.1.4

Raise $9$ to the power of $4$ .

$x \log (81) - \log (6561) = 1$

Step 3

Add $\log (6561)$ to both sides of the equation.

$x \log (81) = 1 + \log (6561)$

Step 4

Divide each term in $x \log (81) = 1 + \log (6561)$ by $\log (81)$ and simplify.

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Step 4.1

Divide each term in $x \log (81) = 1 + \log (6561)$ by $\log (81)$ .

$\frac{x \log (81)}{\log (81)} = \frac{1}{\log (81)} + \frac{\log (6561)}{\log (81)}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $\log (81)$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{x \log (81)}{\log (81)} = \frac{1}{\log (81)} + \frac{\log (6561)}{\log (81)}$

Step 4.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{\log (81)} + \frac{\log (6561)}{\log (81)}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1}{\log (81)} + \frac{\log (6561)}{\log (81)}$

Decimal Form:

$x = 2.52397581 \dots$