Solve each equation. In Exercises 11–34, give irrational solutions as decimals correct to the nearest thousandth. In Exercises 35-40, give solutions in exact form. See Examples 1–4. e^(3x-7) • e^-2x = 4e

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 and are characterized by their unique property that the rate of change is proportional to the function's value. Understanding how to manipulate and solve equations involving exponential functions is crucial for solving the given equation.

Properties of Exponents

The properties of exponents are rules that govern how to simplify and manipulate expressions involving powers. Key properties include the product of powers (a^m * a^n = a^(m+n)), the power of a power ( (a^m)^n = a^(m*n)), and the quotient of powers (a^m / a^n = a^(m-n)). These properties are essential for combining and simplifying the exponential terms in the equation provided.

Solving Exponential Equations

Solving exponential equations often involves isolating the exponential expression and applying logarithms to both sides of the equation. This process allows us to convert the exponential form into a linear form, making it easier to solve for the variable. In this case, recognizing that the equation can be simplified using properties of exponents will lead to a solution that can be expressed in both decimal and exact forms.