Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. e^(x+1)=1/e

Verified step by step guidance

1. Rewrite the right side of the equation $e^\{x+1\}\; =\; \backslash frac\{1\}\{e\}$ as a power of $e$. Recall that $\backslash frac\{1\}\{e\}\; =\; e^\{-1\}$.
2. Now the equation becomes $e^\{x+1\}\; =\; e^\{-1\}$.
3. Since the bases on both sides are the same (both are $e$), set the exponents equal to each other: $x\; +\; 1\; =\; -1$.
4. Solve the resulting linear equation for $x$ by subtracting 1 from both sides: $x\; =\; -1\; -\; 1$.
5. Simplify the right side to find the value of $x$.

Key Concepts

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting expressions so that both sides have the same base, allowing the exponents to be set equal to each other.

Properties of Exponents

Understanding properties such as a^m * a^n = a^(m+n), (a^m)^n = a^(mn), and a^(-n) = 1/a^n is essential. These rules help rewrite expressions and simplify exponential equations to isolate the variable.

Expressing Both Sides with the Same Base

To solve equations like e^(x+1) = 1/e, rewrite both sides using the same base (here, base e). For example, 1/e can be written as e^(-1), enabling you to equate exponents directly and solve for x.