Solve for X log base 4 of X- log base 4 of X-1=1/2

$\log_{4} (X) - \log_{4} (X - 1) = \frac{1}{2}$

Step 1

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

$\log_{4} (\frac{X}{X - 1}) = \frac{1}{2}$

Step 2

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

$4^{\frac{1}{2}} = \frac{X}{X - 1}$

Step 3

Cross multiply to remove the fraction.

$X = 4^{\frac{1}{2}} (X - 1)$

Step 4

Simplify $4^{\frac{1}{2}} (X - 1)$ .

Simplify the expression.

Rewrite $4$ as $2^{2}$ .

$X = {(2^{2})}^{\frac{1}{2}} (X - 1)$

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

$X = 2^{2 (\frac{1}{2})} (X - 1)$

Cancel the common factor of $2$ .

Cancel the common factor.

$X = 2^{2 (\frac{1}{2})} (X - 1)$

Rewrite the expression.

$X = 2^{1} (X - 1)$

Evaluate the exponent.

$X = 2 (X - 1)$

Apply the distributive property.

$X = 2 X + 2 \cdot - 1$

Multiply $2$ by $- 1$ .

$X = 2 X - 2$

Step 5

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

Subtract $2 X$ from both sides of the equation.

$X - 2 X = - 2$

Subtract $2 X$ from $X$ .

$- X = - 2$

Step 6

Divide each term in $- X = - 2$ by $- 1$ and simplify.

Divide each term in $- X = - 2$ by $- 1$ .

$\frac{- X}{- 1} = \frac{- 2}{- 1}$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{X}{1} = \frac{- 2}{- 1}$

Divide $X$ by $1$ .

$X = \frac{- 2}{- 1}$

Simplify the right side.

Divide $- 2$ by $- 1$ .

$X = 2$

Solve for X log base 4 of X- log base 4 of X-1=1/2

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