Solve the equation and express the solution in exact form.
-$\log ( x + 3 ) = 1 - \log x$
A) $\{ 2 \}$
B) $\{ - 5,2 \}$
C) $\{ - 2,5 \}$
D) $\{ - 2 \}$

Question

Quizplus · Accepted Answer

Using the property $\log a + \log b = \log(ab)$, the equation can be rewritten as $\log(x + 3) + \log(x) = 1$, which simplifies to $\log(x(x + 3)) = 1$. Converting to exponential form gives $x(x + 3) = 10$, leading to a quadratic equation $x^2 + 3x - 10 = 0$. Factoring or using the quadratic formula yields $x = 2$ and $x = -5$. However, substituting $x = -5$ back into the original equation results in a logarithm of a negative number, which is undefined. Therefore, the only solution is $x = 2$.

Solve the Equation and Express the Solution in Exact Form log⁡(x+3)=1−log⁡x\log ( x + 3 ) = 1 - \log xlog(x+3)=1−logx

Solve the Equation and Express the Solution in Exact Form $\log ( x + 3 ) = 1 - \log x$