Algebra Examples

$\log (2) + \log (x - 4) = \log (3 x + 8)$

Step 1

Simplify the left side.

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Step 1.1

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

$\log (2 (x - 4)) = \log (3 x + 8)$

Step 1.2

Apply the distributive property.

$\log (2 x + 2 \cdot - 4) = \log (3 x + 8)$

Step 1.3

Multiply $2$ by $- 4$ .

$\log (2 x - 8) = \log (3 x + 8)$

Step 2

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

$2 x - 8 = 3 x + 8$

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 $3 x$ from both sides of the equation.

$2 x - 8 - 3 x = 8$

Step 3.1.2

Subtract $3 x$ from $2 x$ .

$- x - 8 = 8$

Step 3.2

Move all terms not containing $x$ to the right side of the equation.

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Step 3.2.1

Add $8$ to both sides of the equation.

$- x = 8 + 8$

Step 3.2.2

Add $8$ and $8$ .

$- x = 16$

Step 3.3

Divide each term in $- x = 16$ by $- 1$ and simplify.

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Step 3.3.1

Divide each term in $- x = 16$ by $- 1$ .

$\frac{- x}{- 1} = \frac{16}{- 1}$

Step 3.3.2

Simplify the left side.

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Step 3.3.2.1

Dividing two negative values results in a positive value.

$\frac{x}{1} = \frac{16}{- 1}$

Step 3.3.2.2

Divide $x$ by $1$ .

$x = \frac{16}{- 1}$

Step 3.3.3

Simplify the right side.

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Step 3.3.3.1

Divide $16$ by $- 1$ .

$x = - 16$

Step 4

Exclude the solutions that do not make $\log (2) + \log (x - 4) = \log (3 x + 8)$ true.

$No solution$