Solve the equation.
[tex]e^{-x}=\left(e^8\right)^{x+9}[/tex]
[tex]x=[/tex] $\square$ (Type an integer or a simplified fraction.)
Answer (1)
Simplify the equation using exponent rules: e − x = e 8 ( x + 9 ) .
Equate the exponents: − x = 8 ( x + 9 ) .
Distribute and rearrange the equation: − x = 8 x + 72 ⇒ 9 x = − 72 .
Solve for x : x = 9 − 72 = − 8 .
Explanation
Problem Setup We are given the equation e − x = ( e 8 ) x + 9 . Our goal is to solve for x .
Simplifying the Equation First, we can simplify the right side of the equation using the power of a power rule: ( a m ) n = a mn . Applying this rule, we get ( e 8 ) x + 9 = e 8 ( x + 9 ) . So our equation becomes e − x = e 8 ( x + 9 ) .
Equating Exponents Since the bases are equal, we can set the exponents equal to each other: − x = 8 ( x + 9 ) .
Distributing Now, we distribute the 8 on the right side: − x = 8 x + 72 .
Isolating x Next, we want to isolate x . We can add x to both sides of the equation: 0 = 9 x + 72 .
Solving for x Subtract 72 from both sides: − 72 = 9 x .
Final Answer Finally, divide both sides by 9 to solve for x : x = 9 − 72 = − 8 .
Examples
Exponential equations like this appear in various fields, such as calculating the decay of radioactive substances or modeling population growth. For instance, if we have a radioactive substance decaying over time, the amount remaining can be modeled by an exponential equation. Solving for the exponent helps us determine the time it takes for the substance to reach a certain level of decay. Understanding exponential equations is crucial in many scientific and engineering applications.
