Solve for x log base 4 of x+2- log base 4 of x=2

$\log_{4} (x + 2) - \log_{4} (x) = 2$

Step 1

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

$\log_{4} (\frac{x + 2}{x}) = 2$

Step 2

Rewrite $\log_{4} (\frac{x + 2}{x}) = 2$ in exponential form using the definition of a logarithm. If $x$ and $b$ are positive real numbers and $b$ $\neq$ $1$ , then $\log_{b} (x) = y$ is equivalent to $b^{y} = x$ .

$4^{2} = \frac{x + 2}{x}$

Step 3

Cross multiply to remove the fraction.

$x + 2 = 4^{2} (x)$

Step 4

Raise $4$ to the power of $2$ .

$x + 2 = 16 x$

Step 5

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

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

$x + 2 - 16 x = 0$

Subtract $16 x$ from $x$ .

$- 15 x + 2 = 0$

Step 6

Subtract $2$ from both sides of the equation.

$- 15 x = - 2$

Step 7

Divide each term in $- 15 x = - 2$ by $- 15$ and simplify.

Divide each term in $- 15 x = - 2$ by $- 15$ .

$\frac{- 15 x}{- 15} = \frac{- 2}{- 15}$

Simplify the left side.

Cancel the common factor of $- 15$ .

Cancel the common factor.

$\frac{- 15 x}{- 15} = \frac{- 2}{- 15}$

Divide $x$ by $1$ .

$x = \frac{- 2}{- 15}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{2}{15}$

Step 8

The result can be shown in multiple forms.

Exact Form:

$x = \frac{2}{15}$

Decimal Form:

$x = 0.1\overline{3}$

Solve for x log base 4 of x+2- log base 4 of x=2

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