Solve for x log base 25 of 4x+2- log base 25 of 2x-1=1/2

$\log_{25} (4 x + 2) - \log_{25} (2 x - 1) = \frac{1}{2}$

Simplify the left side.

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

$\log_{25} (\frac{4 x + 2}{2 x - 1}) = \frac{1}{2}$

Factor $2$ out of $4 x + 2$ .

Factor $2$ out of $4 x$ .

$\log_{25} (\frac{2 (2 x) + 2}{2 x - 1}) = \frac{1}{2}$

Factor $2$ out of $2$ .

$\log_{25} (\frac{2 (2 x) + 2 (1)}{2 x - 1}) = \frac{1}{2}$

Factor $2$ out of $2 (2 x) + 2 (1)$ .

$\log_{25} (\frac{2 (2 x + 1)}{2 x - 1}) = \frac{1}{2}$

Rewrite $\log_{25} (\frac{2 (2 x + 1)}{2 x - 1}) = \frac{1}{2}$ 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 $b^{y} = x$ .

$25^{\frac{1}{2}} = \frac{2 (2 x + 1)}{2 x - 1}$

Cross multiply to remove the fraction.

$2 (2 x + 1) = 25^{\frac{1}{2}} (2 x - 1)$

Simplify $25^{\frac{1}{2}} (2 x - 1)$ .

Simplify the expression.

Rewrite $25$ as $5^{2}$ .

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

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

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

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Evaluate the exponent.

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

Apply the distributive property.

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

Multiply.

Multiply $2$ by $5$ .

$2 (2 x + 1) = 10 x + 5 \cdot - 1$

Multiply $5$ by $- 1$ .

$2 (2 x + 1) = 10 x - 5$

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

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

$2 (2 x + 1) - 10 x = - 5$

Simplify each term.

Apply the distributive property.

$2 (2 x) + 2 \cdot 1 - 10 x = - 5$

Multiply $2$ by $2$ .

$4 x + 2 \cdot 1 - 10 x = - 5$

Multiply $2$ by $1$ .

$4 x + 2 - 10 x = - 5$

Subtract $10 x$ from $4 x$ .

$- 6 x + 2 = - 5$

Factor $2$ out of $- 6 x + 2$ .

Factor $2$ out of $- 6 x$ .

$2 (- 3 x) + 2 = - 5$

Factor $2$ out of $2$ .

$2 (- 3 x) + 2 (1) = - 5$

Factor $2$ out of $2 (- 3 x) + 2 (1)$ .

$2 (- 3 x + 1) = - 5$

Divide each term in $2 (- 3 x + 1) = - 5$ by $2$ and simplify.

Divide each term in $2 (- 3 x + 1) = - 5$ by $2$ .

$\frac{2 (- 3 x + 1)}{2} = \frac{- 5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 (- 3 x + 1)}{2} = \frac{- 5}{2}$

Divide $- 3 x + 1$ by $1$ .

$- 3 x + 1 = \frac{- 5}{2}$

Simplify the right side.

Move the negative in front of the fraction.

$- 3 x + 1 = - \frac{5}{2}$

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

Subtract $1$ from both sides of the equation.

$- 3 x = - \frac{5}{2} - 1$

To write $- 1$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$- 3 x = - \frac{5}{2} - 1 \cdot \frac{2}{2}$

Combine $- 1$ and $\frac{2}{2}$ .

$- 3 x = - \frac{5}{2} + \frac{- 1 \cdot 2}{2}$

Combine the numerators over the common denominator.

$- 3 x = \frac{- 5 - 1 \cdot 2}{2}$

Simplify the numerator.

Multiply $- 1$ by $2$ .

$- 3 x = \frac{- 5 - 2}{2}$

Subtract $2$ from $- 5$ .

$- 3 x = \frac{- 7}{2}$

Move the negative in front of the fraction.

$- 3 x = - \frac{7}{2}$

Divide each term in $- 3 x = - \frac{7}{2}$ by $- 3$ and simplify.

Divide each term in $- 3 x = - \frac{7}{2}$ by $- 3$ .

$\frac{- 3 x}{- 3} = \frac{- \frac{7}{2}}{- 3}$

Simplify the left side.

Cancel the common factor of $- 3$ .

Cancel the common factor.

$\frac{- 3 x}{- 3} = \frac{- \frac{7}{2}}{- 3}$

Divide $x$ by $1$ .

$x = \frac{- \frac{7}{2}}{- 3}$

Simplify the right side.

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{7}{2} \cdot \frac{1}{- 3}$

Move the negative in front of the fraction.

$x = - \frac{7}{2} (- \frac{1}{3})$

Multiply $- \frac{7}{2} (- \frac{1}{3})$ .

Multiply $- 1$ by $- 1$ .

$x = 1 (\frac{7}{2}) \frac{1}{3}$

Multiply $\frac{7}{2}$ by $1$ .

$x = \frac{7}{2} \cdot \frac{1}{3}$

Multiply $\frac{7}{2}$ by $\frac{1}{3}$ .

$x = \frac{7}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{7}{6}$

The result can be shown in multiple forms.

Exact Form:

$x = \frac{7}{6}$

Decimal Form:

$x = 1.1\overline{6}$

Mixed Number Form:

$x = 1 \frac{1}{6}$

Solve for x log base 25 of 4x+2- log base 25 of 2x-1=1/2

Related Questions

Support LCMGCF.com by disabling your adblocker.