Solve each exponential equation in Exercises 1–22 by expressing each side as a power of the same base and then equating exponents. 4^x=1/√2

Key Concepts

Exponential Equations

Exponential equations are equations in which variables appear as exponents. To solve these equations, one common method is to express both sides of the equation with the same base. This allows us to equate the exponents directly, simplifying the process of finding the variable's value.

Properties of Exponents

The properties of exponents, such as the product of powers, power of a power, and the power of a product, are essential for manipulating exponential expressions. For instance, the property that states a^(m/n) = n√(a^m) helps in rewriting expressions like 1/√2 in terms of powers, facilitating the comparison of bases.

Equating Exponents

When two exponential expressions with the same base are equal, their exponents must also be equal. This principle is crucial for solving exponential equations. By transforming both sides of the equation to have a common base, we can set the exponents equal to each other, leading to a straightforward algebraic solution.