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

$\log_{2} (x + 5) - \log_{2} (2 x - 1) = 5$

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

$\log_{2} (\frac{x + 5}{2 x - 1}) = 5$

Rewrite $\log_{2} (\frac{x + 5}{2 x - 1}) = 5$ 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 $b^{y} = x$ .

$2^{5} = \frac{x + 5}{2 x - 1}$

Cross multiply to remove the fraction.

$x + 5 = 2^{5} (2 x - 1)$

Simplify $2^{5} (2 x - 1)$ .

Raise $2$ to the power of $5$ .

$x + 5 = 32 (2 x - 1)$

Apply the distributive property.

$x + 5 = 32 (2 x) + 32 \cdot - 1$

Multiply.

Multiply $2$ by $32$ .

$x + 5 = 64 x + 32 \cdot - 1$

Multiply $32$ by $- 1$ .

$x + 5 = 64 x - 32$

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

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

$x + 5 - 64 x = - 32$

Subtract $64 x$ from $x$ .

$- 63 x + 5 = - 32$

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

Subtract $5$ from both sides of the equation.

$- 63 x = - 32 - 5$

Subtract $5$ from $- 32$ .

$- 63 x = - 37$

Divide each term in $- 63 x = - 37$ by $- 63$ and simplify.

Divide each term in $- 63 x = - 37$ by $- 63$ .

$\frac{- 63 x}{- 63} = \frac{- 37}{- 63}$

Simplify the left side.

Cancel the common factor of $- 63$ .

Cancel the common factor.

$\frac{- 63 x}{- 63} = \frac{- 37}{- 63}$

Divide $x$ by $1$ .

$x = \frac{- 37}{- 63}$

Simplify the right side.

Dividing two negative values results in a positive value.

$x = \frac{37}{63}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{37}{63}$

Decimal Form:

$x = 0.\overline{587301}$

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

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