Solve each equation. See Examples 4–6. (1/e)^-x = (1/e^2)^(x+1)

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of 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 and are fundamental in solving equations involving exponents, as they allow us to manipulate and equate powers effectively.

Properties of Exponents

The properties of exponents are rules that govern how to manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the quotient of powers (a^m / a^n = a^(m-n)), and the power of a power ( (a^m)^n = a^(m*n)). Understanding these properties is essential for simplifying and solving exponential equations.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for exponents in equations. The logarithm log_b(a) answers the question: 'To what exponent must the base b be raised to produce a?' This concept is crucial when dealing with equations where the variable is in the exponent, as it enables us to isolate the variable and find its value.