Solve each equation. See Examples 4–6. (√2)^(x+4) = 4^x

Verified step by step guidance

1. Identify the bases and rewrite the equation using the same base if possible. Here, recognize that 4 can be written as \((2^2)\), so rewrite the equation as \((\sqrt{2})^{x+4} = (2^2)^x\).
2. Simplify the bases. Recall that \(\sqrt{2} = 2^{1/2}\), so the equation becomes \((2^{1/2})^{x+4} = (2^2)^x\).
3. Apply the power of a power property, which states that \((a^m)^n = a^{m \cdot n}\). Thus, the equation simplifies to \$2^{(1/2)(x+4)} = 2^{2x}$.
4. Since the bases are the same, set the exponents equal to each other: \((1/2)(x+4) = 2x\).
5. Solve the resulting linear equation for \(x\). First, distribute the \$1/2\( across \)(x+4)\(, then combine like terms and isolate \)x$ on one side of the equation.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a^x, where 'a' is a positive constant. These functions exhibit rapid growth or decay and are characterized by their constant base raised to a variable exponent. Understanding the properties of exponential functions is crucial for solving equations involving them, as they can often be manipulated using logarithms.

Logarithms

Logarithms are the inverse operations of exponentiation, allowing us to solve for the exponent in equations of the form a^x = b. The logarithm log_a(b) answers the question: 'To what exponent must the base 'a' be raised to produce 'b'?' Familiarity with logarithmic properties, such as the product, quotient, and power rules, is essential for simplifying and solving exponential equations.

Change of Base Formula

The change of base formula allows us to convert logarithms from one base to another, which is particularly useful when dealing with different bases in exponential equations. The formula states that log_a(b) = log_c(b) / log_c(a) for any positive base 'c'. This concept is important for solving equations where the bases of the exponentials differ, enabling us to express them in a common logarithmic form.