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

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form f(x) = a * b^x, where 'a' is a constant, 'b' is the base (a positive real number), and 'x' is the exponent. These functions exhibit rapid growth or decay, depending on the base. Understanding their properties, such as the behavior of the function as 'x' approaches positive or negative infinity, is crucial for solving exponential equations.

Equating Exponents

When two exponential expressions with the same base are equal, their exponents must also be equal. This principle allows us to simplify and solve equations by transforming them into a linear form. For example, if a^m = a^n, then m = n, provided 'a' is not zero or one. This concept is essential for solving the given equation by expressing both sides with a common base.

Properties of Exponents

The properties of exponents, such as the product of powers, quotient of powers, and power of a power, provide rules for manipulating exponential expressions. For instance, a^m * a^n = a^(m+n) and a^m / a^n = a^(m-n). These properties are vital for rewriting expressions in the same base, which is necessary for solving exponential equations like the one presented.