Solve for x log of x+4- log of 2 = log of 5x+1

$\log (x + 4) - \log (2) = \log (5 x + 1)$

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

$\log (\frac{x + 4}{2}) = \log (5 x + 1)$

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

$\frac{x + 4}{2} = 5 x + 1$

Solve for $x$ .

Multiply both sides by $2$ .

$\frac{x + 4}{2} \cdot 2 = (5 x + 1) \cdot 2$

Simplify.

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{x + 4}{2} \cdot 2 = (5 x + 1) \cdot 2$

Rewrite the expression.

$x + 4 = (5 x + 1) \cdot 2$

Simplify the right side.

Simplify $(5 x + 1) \cdot 2$ .

Apply the distributive property.

$x + 4 = 5 x \cdot 2 + 1 \cdot 2$

Multiply.

Multiply $2$ by $5$ .

$x + 4 = 10 x + 1 \cdot 2$

Multiply $2$ by $1$ .

$x + 4 = 10 x + 2$

Solve for $x$ .

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

Subtract $10 x$ from both sides of the equation.

$x + 4 - 10 x = 2$

Subtract $10 x$ from $x$ .

$- 9 x + 4 = 2$

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

Subtract $4$ from both sides of the equation.

$- 9 x = 2 - 4$

Subtract $4$ from $2$ .

$- 9 x = - 2$

Divide each term in $- 9 x = - 2$ by $- 9$ and simplify.

Divide each term in $- 9 x = - 2$ by $- 9$ .

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

Simplify the left side.

Cancel the common factor of $- 9$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{2}{9}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{9}$

Decimal Form:

$x = 0.\overline{2}$

Solve for x log of x+4- log of 2 = log of 5x+1

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