Solve each equation. See Examples 4–6. 4^(x-2) = 2^(3x+3)

Verified step by step guidance

1. Recognize that both sides of the equation have bases that are powers of 2. Rewrite 4 as 2^2, so the equation becomes (2^2)^(x-2) = 2^(3x+3).
2. Apply the power of a power property of exponents, which states (a^m)^n = a^(m*n). This gives us 2^(2(x-2)) = 2^(3x+3).
3. Since the bases are the same, set the exponents equal to each other: 2(x-2) = 3x + 3.
4. Distribute the 2 on the left side: 2x - 4 = 3x + 3.
5. Solve for x by isolating x on one side of the equation.

Key Concepts

Exponential Equations

Exponential equations involve variables in the exponent and can often be solved by rewriting them in a common base. Understanding how to manipulate exponents and apply properties of exponents is crucial for solving these types of equations.

Properties of Exponents

The properties of exponents, such as the product of powers, power of a power, and quotient of powers, allow us to simplify expressions involving exponents. For example, knowing that 4 can be expressed as 2^2 helps in rewriting the equation to a common base, facilitating easier solving.

Logarithms

Logarithms are the inverse operations of exponentiation and are useful for solving equations where the variable is in the exponent. Understanding how to apply logarithmic properties can help isolate the variable and find its value in exponential equations.