Solve each equation. See Examples 4–6. 4^x = 2

Verified step by step guidance

1. Recognize that the equation \$4^x = 2$ involves exponential expressions with different bases. To solve it, try to express both sides with the same base if possible.
2. Rewrite the base 4 as a power of 2, since \$4 = 2^2\(. So, rewrite the equation as \)(2^2)^x = 2$.
3. Use the power of a power property: \((a^m)^n = a^{m \cdot n}\). This gives \$2^{2x} = 2^1$.
4. Since the bases are the same (both base 2), set the exponents equal to each other: \$2x = 1$.
5. Solve the resulting linear equation for \(x\) by dividing both sides by 2: \(x = \frac{1}{2}\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position, such as 4^x = 2. Solving these requires understanding how to manipulate and rewrite expressions to isolate the variable in the exponent.

Properties of Exponents

Properties of exponents allow rewriting expressions with the same base or converting bases to a common base. For example, 4 can be written as 2^2, which helps in equating exponents when bases match.

Logarithms

Logarithms are the inverse operations of exponentials and are used to solve equations where the variable is an exponent. Applying logarithms helps isolate the exponent and solve for the variable when bases cannot be easily matched.