Problem 22 - College Algebra Essentials

Question

Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$e^{x+4}=\frac{1}{e^{2 x}}$$

Step-by-Step Solution

Answer

The solution to the given exponential equation is \( x = -\frac{4}{3} \).

Step 1: Rewrite fraction as negative exponent

Remember that any fraction with the denominator \( e^{2x} \) can be written as \( e^{-2x} \). So rewrite \( e^{x+4} = \frac{1}{e^{2x}} \) as \( e^{x+4} = e^{-2x} \).

Step 2: Equate the exponents

Now that both sides are expressed as powers of the base \( e \), equate the exponents to each other. This gives you the equation \( x+4 = -2x \).

Step 3: Solve for x

Rearrange this equation to find \( x \). First, add \( 2x \) on both sides to give \( 3x+4 = 0 \), then subtract \( 4 \) on both sides to get \( 3x = -4 \). Finally, divide the equation by \( 3 \) to solve for \( x \), resulting in \( x = -\frac{4}{3} \).