Solve for x 2 natural log of x- natural log of 2x-9 = natural log of 7x- natural log of x-2

$2 \ln (x) - \ln (2 x - 9) = \ln (7 x) - \ln (x - 2)$

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

$2 \ln (x) - \ln (2 x - 9) = \ln (\frac{7 x}{x - 2})$

Move all the terms containing a logarithm to the left side of the equation.

$2 \ln (x) - \ln (2 x - 9) - \ln (\frac{7 x}{x - 2}) = 0$

Simplify the left side.

Simplify $2 \ln (x) - \ln (2 x - 9) - \ln (\frac{7 x}{x - 2})$ .

Simplify $2 \ln (x)$ by moving $2$ inside the logarithm.

$\ln (x^{2}) - \ln (2 x - 9) - \ln (\frac{7 x}{x - 2}) = 0$

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

$\ln (\frac{x^{2}}{2 x - 9}) - \ln (\frac{7 x}{x - 2}) = 0$

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

$\ln (\frac{\frac{x^{2}}{2 x - 9}}{\frac{7 x}{x - 2}}) = 0$

Multiply the numerator by the reciprocal of the denominator.

$\ln (\frac{x^{2}}{2 x - 9} \cdot \frac{x - 2}{7 x}) = 0$

Cancel the common factor of $x$ .

Factor $x$ out of $x^{2}$ .

$\ln (\frac{x \cdot x}{2 x - 9} \cdot \frac{x - 2}{7 x}) = 0$

Factor $x$ out of $7 x$ .

$\ln (\frac{x \cdot x}{2 x - 9} \cdot \frac{x - 2}{x \cdot 7}) = 0$

Cancel the common factor.

$\ln (\frac{x \cdot x}{2 x - 9} \cdot \frac{x - 2}{x \cdot 7}) = 0$

Rewrite the expression.

$\ln (\frac{x}{2 x - 9} \cdot \frac{x - 2}{7}) = 0$

Multiply $\frac{x}{2 x - 9}$ by $\frac{x - 2}{7}$ .

$\ln (\frac{x (x - 2)}{(2 x - 9) \cdot 7}) = 0$

Move $7$ to the left of $2 x - 9$ .

$\ln (\frac{x (x - 2)}{7 (2 x - 9)}) = 0$

Rewrite $\ln (\frac{x (x - 2)}{7 (2 x - 9)}) = 0$ 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 $b^{y} = x$ .

$e^{0} = \frac{x (x - 2)}{7 (2 x - 9)}$

Cross multiply to remove the fraction.

$x (x - 2) = e^{0} (7 (2 x - 9))$

Simplify $e^{0} (7 (2 x - 9))$ .

Simplify the expression.

Anything raised to $0$ is $1$ .

$x (x - 2) = 1 (7 (2 x - 9))$

Multiply $7 (2 x - 9)$ by $1$ .

$x (x - 2) = 7 (2 x - 9)$

Apply the distributive property.

$x (x - 2) = 7 (2 x) + 7 \cdot - 9$

Multiply.

Multiply $2$ by $7$ .

$x (x - 2) = 14 x + 7 \cdot - 9$

Multiply $7$ by $- 9$ .

$x (x - 2) = 14 x - 63$

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

Subtract $14 x$ from both sides of the equation.

$x (x - 2) - 14 x = - 63$

Simplify each term.

Apply the distributive property.

$x \cdot x + x \cdot - 2 - 14 x = - 63$

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 2 - 14 x = - 63$

Move $- 2$ to the left of $x$ .

$x^{2} - 2 x - 14 x = - 63$

Subtract $14 x$ from $- 2 x$ .

$x^{2} - 16 x = - 63$

Factor $x$ out of $x^{2} - 16 x$ .

Factor $x$ out of $x^{2}$ .

$x \cdot x - 16 x = - 63$

Factor $x$ out of $- 16 x$ .

$x \cdot x + x \cdot - 16 = - 63$

Factor $x$ out of $x \cdot x + x \cdot - 16$ .

$x (x - 16) = - 63$

Simplify $x (x - 16)$ .

Apply the distributive property.

$x \cdot x + x \cdot - 16 = - 63$

Simplify the expression.

Multiply $x$ by $x$ .

$x^{2} + x \cdot - 16 = - 63$

Move $- 16$ to the left of $x$ .

$x^{2} - 16 x = - 63$

Add $63$ to both sides of the equation.

$x^{2} - 16 x + 63 = 0$

Factor $x^{2} - 16 x + 63$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $63$ and whose sum is $- 16$ .

$- 9, - 7$

Write the factored form using these integers.

$(x - 9) (x - 7) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 9 = 0$

$x - 7 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Set $x - 7$ equal to $0$ and solve for $x$ .

Set $x - 7$ equal to $0$ .

$x - 7 = 0$

Add $7$ to both sides of the equation.

$x = 7$

The final solution is all the values that make $(x - 9) (x - 7) = 0$ true.

$x = 9, 7$

Solve for x 2 natural log of x- natural log of 2x-9 = natural log of 7x- natural log of x-2

Related Questions

Support LCMGCF.com by disabling your adblocker.