Solve for x natural log of 10- natural log of x-3 = natural log of x

$\ln (10) - \ln (x - 3) = \ln (x)$

Step 1

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

$\ln (\frac{10}{x - 3}) = \ln (x)$

Step 2

For the equation to be equal, the argument of the logarithms on both sides of the equation must be equal.

$\frac{10}{x - 3} = x$

Step 3

Solve for $x$ .

Find the LCD of the terms in the equation.

Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x - 3, 1$

Remove parentheses.

$x - 3, 1$

The LCM of one and any expression is the expression.

$x - 3$

Multiply each term in $\frac{10}{x - 3} = x$ by $x - 3$ to eliminate the fractions.

Multiply each term in $\frac{10}{x - 3} = x$ by $x - 3$ .

$\frac{10}{x - 3} (x - 3) = x (x - 3)$

Simplify the left side.

Cancel the common factor of $x - 3$ .

Cancel the common factor.

$\frac{10}{x - 3} (x - 3) = x (x - 3)$

Rewrite the expression.

$10 = x (x - 3)$

Simplify the right side.

Apply the distributive property.

$10 = x \cdot x + x \cdot - 3$

Simplify the expression.

Multiply $x$ by $x$ .

$10 = x^{2} + x \cdot - 3$

Move $- 3$ to the left of $x$ .

$10 = x^{2} - 3 x$

Solve the equation.

Rewrite the equation as $x^{2} - 3 x = 10$ .

$x^{2} - 3 x = 10$

Subtract $10$ from both sides of the equation.

$x^{2} - 3 x - 10 = 0$

Factor $x^{2} - 3 x - 10$ 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 $- 10$ and whose sum is $- 3$ .

$- 5, 2$

Write the factored form using these integers.

$(x - 5) (x + 2) = 0$

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

$x - 5 = 0$

$x + 2 = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

The final solution is all the values that make $(x - 5) (x + 2) = 0$ true.

$x = 5, - 2$

Step 4

Exclude the solutions that do not make $\ln (10) - \ln (x - 3) = \ln (x)$ true.

$x = 5$

Solve for x natural log of 10- natural log of x-3 = natural log of x

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