Solve the logarithmic equation.
-$\log x = 1 - \log ( x + 3 )$
A) $\{ 1 , - 3 \}$
B) $\{ 2 , - 5 \}$
C) $\left\{ \frac { 3 } { 2 } , 5 \right\}$
D) $\{ 2 \}$

Question

Quizplus · Accepted Answer

To solve the equation $\log x = 1 - \log ( x + 3 )$, we can first combine the logarithms on one side: $\log x + \log ( x + 3 ) = 1$. This can be rewritten using the logarithm product rule as $\log [x(x + 3)] = 1$. Converting from logarithmic to exponential form gives $10^1 = x(x + 3)$, which simplifies to $x^2 + 3x - 10 = 0$. Factoring or using the quadratic formula yields $x = 2$ and $x = -5$. However, because $\log x$ and $\log (x + 3)$ are only defined for positive arguments, $x = -5$ is not a valid solution since it would make $x + 3 = -2$, which is not allowed in the logarithm. Therefore, the only solution is $x = 2$.

Solve the Logarithmic Equation log⁡x=1−log⁡(x+3)\log x = 1 - \log ( x + 3 )logx=1−log(x+3)

Solve the Logarithmic Equation $\log x = 1 - \log ( x + 3 )$