Problem 21 - College Algebra Essentials

Question

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

Step-by-Step Solution

Answer

The solution of the equation \(e^{x+1}=\frac{1}{e}\) is \(x=-2\).

Step 1: Convert the right side of the equation to the same base

The equation given is \(e^{x+1}=\frac{1}{e}\). The base on the right side of the equation can be rewritten by recalling that the reciprocal of a number \(a\) is \(a^{-1}\). So, \(\frac{1}{e}\) can be rewritten as \(e^{-1}\).

Step 2: Equating exponents

After changing the base of the right-hand side to be the same as the base on the left-hand side, the equation is \(e^{x+1}=e^{-1}\). Now, since the bases are equal, the exponents must also be equal so the equation becomes \(x+1=-1\).

Step 3: Solve for x

Solve \(x+1=-1\) for \(x\) would be \(x=-1-1\) which simplifies to \(x=-2\).