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

Verified step by step guidance

1. Rewrite the equation \(4^{x} = \frac{1}{\sqrt{2}}\) by expressing both sides as powers of the same base. Note that 4 can be written as \$2^{2}$ and \(\sqrt{2}\) can be written as \(2^{\frac{1}{2}}\).
2. Rewrite the left side as \(\left(2^{2}\right)^{x}\) and the right side as \(\frac{1}{2^{\frac{1}{2}}}\).
3. Use the property of exponents \(\left(a^{m}\right)^{n} = a^{mn}\) to simplify the left side to \$2^{2x}$.
4. Rewrite the right side \(\frac{1}{2^{\frac{1}{2}}}\) as \(2^{-\frac{1}{2}}\) using the negative exponent rule \(\frac{1}{a^{m}} = a^{-m}\).
5. Since the bases are the same (base 2), set the exponents equal: \(2x = -\frac{1}{2}\). Then solve for \(x\) by dividing both sides by 2.

Key Concepts

Exponential Equations

An exponential equation is one in which variables appear as exponents. Solving these equations often involves rewriting both sides with the same base so that the exponents can be set equal, simplifying the problem to solving a linear equation.

Expressing Numbers as Powers of the Same Base

To solve exponential equations, it is essential to rewrite each side as powers of the same base. For example, 1/√2 can be expressed as 2 raised to a negative fractional exponent, allowing the equation to be compared directly by their exponents.

Equating Exponents

Once both sides of an exponential equation have the same base, the exponents can be set equal to each other. This step transforms the problem into a simpler algebraic equation, which can be solved using standard algebraic methods.