Precalculus Examples

$\ln (4) + \ln (x) = \ln (8) + \ln (2 x - 4)$

Step 1

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

$\ln (4 x) = \ln (8) + \ln (2 x - 4)$

Step 2

Simplify the right side.

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Step 2.1

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

$\ln (4 x) = \ln (8 (2 x - 4))$

Step 2.2

Apply the distributive property.

$\ln (4 x) = \ln (8 (2 x) + 8 \cdot - 4)$

Step 2.3

Multiply.

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Step 2.3.1

Multiply $2$ by $8$ .

$\ln (4 x) = \ln (16 x + 8 \cdot - 4)$

Step 2.3.2

Multiply $8$ by $- 4$ .

$\ln (4 x) = \ln (16 x - 32)$

Step 3

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

$4 x = 16 x - 32$

Step 4

Solve for $x$ .

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Step 4.1

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

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Step 4.1.1

Subtract $16 x$ from both sides of the equation.

$4 x - 16 x = - 32$

Step 4.1.2

Subtract $16 x$ from $4 x$ .

$- 12 x = - 32$

Step 4.2

Divide each term in $- 12 x = - 32$ by $- 12$ and simplify.

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Step 4.2.1

Divide each term in $- 12 x = - 32$ by $- 12$ .

$\frac{- 12 x}{- 12} = \frac{- 32}{- 12}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $- 12$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{- 12 x}{- 12} = \frac{- 32}{- 12}$

Step 4.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 32}{- 12}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Cancel the common factor of $- 32$ and $- 12$ .

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Step 4.2.3.1.1

Factor $- 4$ out of $- 32$ .

$x = \frac{- 4 (8)}{- 12}$

Step 4.2.3.1.2

Cancel the common factors.

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Step 4.2.3.1.2.1

Factor $- 4$ out of $- 12$ .

$x = \frac{- 4 \cdot 8}{- 4 \cdot 3}$

Step 4.2.3.1.2.2

Cancel the common factor.

$x = \frac{- 4 \cdot 8}{- 4 \cdot 3}$

Step 4.2.3.1.2.3

Rewrite the expression.

$x = \frac{8}{3}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{8}{3}$

Decimal Form:

$x = 2.\overline{6}$

Mixed Number Form:

$x = 2 \frac{2}{3}$