Questions: Find the exact solution(s) to the equation: e^(3 x)=(1/e^2)^(3 x+1) x=

Transcript text: Find the exact solution(s) to the equation: $e^{3 x}=\left(\frac{1}{e^{2}}\right)^{3 x+1}$ \[ x= \]

Solution

Solution Steps

To solve the equation $ e^{3x} = \left(\frac{1}{e^2}\right)^{3x+1} $, we can start by simplifying the right-hand side. The expression $\left(\frac{1}{e^2}\right)^{3x+1}$ can be rewritten as $e^{-2(3x+1)}$. Then, we equate the exponents of the base $e$ on both sides of the equation and solve for $x$.

Step 1: Rewrite the Equation

We start with the equation: \[ e^{3x} = \left(\frac{1}{e^2}\right)^{3x+1} \] This can be rewritten as: \[ e^{3x} = e^{-2(3x + 1)} \]

Step 2: Simplify the Right-Hand Side

Next, we simplify the right-hand side: \[ e^{3x} = e^{-6x - 2} \]

Step 3: Set the Exponents Equal

Since the bases are the same, we can set the exponents equal to each other: \[ 3x = -6x - 2 \]

Step 4: Solve for $x$

Now, we solve for $x$: \[ 3x + 6x = -2 \implies 9x = -2 \implies x = -\frac{2}{9} \]