Problem 100 - Algebra and Trigonometry

Question

Solve each equation. $$ \ln 3-\ln (x+5)-\ln x=0 $$

Step-by-Step Solution

Answer

The solution for the equation is \(x = 1\).

Step 1: Combine logarithms

Utilize the subtraction property of logarithms to combine them: \( \ln 3 - \ln(x+5) - \ln x = \ln \left( \frac{3}{(x+5)x} \right) = 0 \)

Step 2: Use definition of logarithm

Recall that \(\ln a = b\) is equivalent to \(e^b = a\). Thus, the equation becomes: \( e^0 = \frac{3}{(x+5)x} \)

Step 3: Simplify and solve for x

Solving \( e^0 = \frac{3}{(x+5)x} \), we get \(1 = \frac{3}{(x+5)x}\). Clearing the fraction gives \(x^2 + 5x - 3 = 0\). Factoring this quadratic equation gives \((x+3)(x-1) = 0\). Therefore, the possible solutions are \(x = -3\) and \(x = 1\)

Step 4: Check validity of the solutions

We can't use \(x = -3\) in the original equation because it would result in a negative argument to the natural logarithm, which is undefined. Therefore, the only valid solution is \(x = 1\) .