Solve the given exponential equation. Round answer to three decimal places.
-$5 ^ { 2 x } + 5 ( x + 1 ) - 24 = 0$
A) {0.683}
B) {1.147}
C) {1.099}
D) {1.292}

Question

Quizplus · Accepted Answer

This equation can be solved by recognizing it as a quadratic equation in disguise. Let $y = 5^x$, then the equation becomes $y^2 + 5y - 24 = 0$. Solving this quadratic equation gives $y = 3$ or $y = -8$. Since $y = 5^x$ and $5^x > 0$ for any real number $x$, the solution $y = -8$ is not valid. Therefore, we have $5^x = 3$. Taking the logarithm of both sides gives $x \log(5) = \log(3)$, and solving for $x$ gives $x = \log(3) / \log(5)$, which approximately equals 0.683.

Solve the Given Exponential Equation 52x+5(x+1)−24=05 ^ { 2 x } + 5 ( x + 1 ) - 24 = 052x+5(x+1)−24=0

Solve the Given Exponential Equation $5 ^ { 2 x } + 5 ( x + 1 ) - 24 = 0$