ln 10 − ln(7 − 𝑥) = ln 𝑥

Sure, let's solve the equation step by step.

The equation is: ln 10 - ln(7 - x) = ln x

Step 1: Use the properties of logarithms to simplify the equation. The difference of two logarithms is the logarithm of the quotient, so we can rewrite the equation as:

ln(10/(7 - x)) = ln x

Step 2: If the natural logarithms of two numbers are equal, then the numbers themselves are equal. So we can write:

10/(7 - x) = x

Step 3: Solve this equation for x. First, multiply both sides by (7 - x) to get rid of the denominator:

10 = x*(7 - x)

Step 4: Distribute x on the right side:

10 = 7x - x^2

Step 5: Rearrange the equation to standard quadratic form:

x^2 - 7x + 10 = 0

Step 6: Factor the quadratic equation:

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

Step 7: Set each factor equal to zero and solve for x:

x - 2 = 0 => x = 2

x - 5 = 0 => x = 5

So the solutions to the equation are x = 2 and x = 5. However, we need to check these solutions in the original equation because logarithms are not defined for negative numbers or zero.

For x = 2, ln(10/(7 - 2)) = ln 2, which is true.

For x = 5, ln(10/(7 - 5)) = ln 5, which is also true.

So both x = 2 and x = 5 are solutions to the equation.