Solve the logarithmic equation. Be sure to reject any value that is not in the domain of the original logarithmic
expressions. Give the exact answer.
-$\log x + \log ( x + 1 ) = \log 72$
A) $\{ 8 \}$
B) $\{ 9 \}$
C) $\{ 9 , - 8 \}$
D) $\left\{ \frac { 73 } { 2 } \right\}$

Question

Quizplus · Accepted Answer

Using the property $\log a + \log b = \log(ab)$, the equation becomes $\log(x(x + 1)) = \log 72$. Solving $x(x + 1) = 72$ gives $x^2 + x - 72 = 0$. Factoring, we get $(x + 9)(x - 8) = 0$, so $x = -9$ or $x = 8$. However, $-9$ is not in the domain of the original logarithmic expressions because $\log x$ and $\log(x + 1)$ are undefined for $x \leq 0$. Thus, the only solution is $x = 8$.

Solve the Logarithmic Equation log⁡x+log⁡(x+1)=log⁡72\log x + \log ( x + 1 ) = \log 72logx+log(x+1)=log72

Solve the Logarithmic Equation $\log x + \log ( x + 1 ) = \log 72$