Solve for x log base 4 of x+1- log base 4 of x-2=2 log base 4 of 8

$\log_{4} (x + 1) - \log_{4} (x - 2) = 2 \log_{4} (8)$

Step 1

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

$\log_{4} (\frac{x + 1}{x - 2}) = 2 \log_{4} (8)$

Step 2

Simplify the right side.

Logarithm base $4$ of $8$ is $\frac{3}{2}$ .

$\log_{4} (\frac{x + 1}{x - 2}) = 2 (\frac{3}{2})$

Cancel the common factor of $2$ .

Cancel the common factor.

$\log_{4} (\frac{x + 1}{x - 2}) = 2 (\frac{3}{2})$

Rewrite the expression.

$\log_{4} (\frac{x + 1}{x - 2}) = 3$

Step 3

Rewrite $\log_{4} (\frac{x + 1}{x - 2}) = 3$ 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 $b^{y} = x$ .

$4^{3} = \frac{x + 1}{x - 2}$

Step 4

Cross multiply to remove the fraction.

$x + 1 = 4^{3} (x - 2)$

Step 5

Simplify $4^{3} (x - 2)$ .

Raise $4$ to the power of $3$ .

$x + 1 = 64 (x - 2)$

Apply the distributive property.

$x + 1 = 64 x + 64 \cdot - 2$

Multiply $64$ by $- 2$ .

$x + 1 = 64 x - 128$

Step 6

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

Subtract $64 x$ from both sides of the equation.

$x + 1 - 64 x = - 128$

Subtract $64 x$ from $x$ .

$- 63 x + 1 = - 128$

Step 7

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

Subtract $1$ from both sides of the equation.

$- 63 x = - 128 - 1$

Subtract $1$ from $- 128$ .

$- 63 x = - 129$

Step 8

Divide each term in $- 63 x = - 129$ by $- 63$ and simplify.

Divide each term in $- 63 x = - 129$ by $- 63$ .

$\frac{- 63 x}{- 63} = \frac{- 129}{- 63}$

Simplify the left side.

Cancel the common factor of $- 63$ .

Cancel the common factor.

$\frac{- 63 x}{- 63} = \frac{- 129}{- 63}$

Divide $x$ by $1$ .

$x = \frac{- 129}{- 63}$

Simplify the right side.

Cancel the common factor of $- 129$ and $- 63$ .

Factor $- 3$ out of $- 129$ .

$x = \frac{- 3 (43)}{- 63}$

Cancel the common factors.

Factor $- 3$ out of $- 63$ .

$x = \frac{- 3 \cdot 43}{- 3 \cdot 21}$

Cancel the common factor.

$x = \frac{- 3 \cdot 43}{- 3 \cdot 21}$

Rewrite the expression.

$x = \frac{43}{21}$

Step 9

The result can be shown in multiple forms.

Exact Form:

$x = \frac{43}{21}$

Decimal Form:

$x = 2.\overline{047619}$

Mixed Number Form:

$x = 2 \frac{1}{21}$

Solve for x log base 4 of x+1- log base 4 of x-2=2 log base 4 of 8

Related Questions

Support LCMGCF.com by disabling your adblocker.