Solve each equation. 4^x-2 = 2^3x+3

Verified step by step guidance

1. Recognize that the bases on both sides of the equation can be expressed as powers of the same base. Since 4 and 2 are related by 4 = 2^2, rewrite 4^(x-2) as (2^2)^(x-2).
2. Apply the power of a power property: (a^m)^n = a^{m \cdot n}. So, (2^2)^{x-2} becomes 2^{2(x-2)}.
3. Rewrite the equation with the same base: 2^{2(x-2)} = 2^{3x+3}.
4. Since the bases are the same and the expressions are equal, set the exponents equal to each other: 2(x-2) = 3x + 3.
5. Solve the resulting linear equation for x by expanding and isolating x: 2x - 4 = 3x + 3.

Key Concepts

Exponential Equations

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

Properties of Exponents

Properties of exponents, such as the product, quotient, and power rules, allow simplification and rewriting of exponential expressions. For example, expressing 4 as 2 squared helps rewrite 4^(x-2) as (2^2)^(x-2) = 2^(2x-4), enabling comparison of exponents with the same base.

Equating Exponents

When exponential expressions have the same base and are set equal, their exponents must be equal. This principle allows converting an exponential equation into a linear equation in terms of the variable, which can then be solved using algebraic methods.