Find all the real-number roots of the equation. Give an exact expression for the root and also(where appropriate) a calculator approximation rounded to three decimal places. $\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0$
A) $x = \frac { - 9 + \sqrt { 85 } } { 2 } \approx 0.110$
B) $x = \frac { - 9 - \sqrt { 85 } } { 2 } \approx - 9.110$
C) $x = \frac { 9 - \sqrt { 85 } } { 2 } \approx - 0.253$
D) $x = \frac { 9 + \sqrt { 85 } } { 2 } \approx 0.253$
E) no solution

Question

Quizplus · Accepted Answer

To solve the equation $\log_7 x + \log_7 (x + 9) = 0$, we can use the property of logarithms that allows us to combine the two logs into a single log: $\log_7(x(x + 9)) = 0$. This simplifies to $x(x + 9) = 7^0 = 1$, leading to the quadratic equation $x^2 + 9x - 1 = 0$. Using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, with $a = 1$, $b = 9$, and $c = -1$, we find $x = \frac{-9 \pm \sqrt{85}}{2}$. This gives two roots, but only $x = \frac{-9 + \sqrt{85}}{2}$ is a real number greater than 0, which is a requirement for the logarithm's argument. The approximation of this root is $0.110$.

Find All the Real-Number Roots of the Equation log⁡7x+log⁡7(x+9)=0\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0log7​x+log7​(x+9)=0

Find All the Real-Number Roots of the Equation $\log _ { 7 } x + \log _ { 7 } ( x + 9 ) = 0$