Solve each equation. (√2)^x+4 = 4^x

Verified step by step guidance

1. Rewrite both sides of the equation with the same base or express them in terms of powers with a common base. Note that \(\sqrt{2}\) can be written as \(2^{\frac{1}{2}}\) and \(4\) can be written as \$2^2$.
2. Express the equation \((\sqrt{2})^{x+4} = 4^x\) as \(\left(2^{\frac{1}{2}}\right)^{x+4} = \left(2^2\right)^x\).
3. Use the power of a power property \(\left(a^m\right)^n = a^{mn}\) to simplify both sides: \(2^{\frac{1}{2}(x+4)} = 2^{2x}\).
4. Since the bases are the same and both sides are equal, set the exponents equal to each other: \(\frac{1}{2}(x+4) = 2x\).
5. Solve the resulting linear equation for \(x\) by first multiplying both sides by 2 to clear the fraction, then isolating \(x\).

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent position. Solving these requires rewriting expressions with a common base or using logarithms to isolate the variable. Understanding how to manipulate exponents is essential for finding the value of the unknown.

Properties of Exponents

Properties of exponents, such as the product rule, power rule, and rewriting radicals as fractional exponents, allow simplification of expressions. For example, √2 can be written as 2^(1/2), enabling easier comparison and manipulation of terms with the same base.

Logarithms

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