Solve each equation. x = 2^log2 9

Verified step by step guidance

1. Recognize that the equation is in the form \( x = 2^{\log_2 9} \).
2. Use the property of logarithms and exponents: \( a^{\log_a b} = b \).
3. Apply this property to the equation: \( 2^{\log_2 9} = 9 \).
4. Conclude that \( x = 9 \) based on the property.
5. Verify the solution by substituting back into the original equation to ensure consistency.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions in the form of f(x) = a^x, where 'a' is a positive constant. They are characterized by their rapid growth or decay and are fundamental in various applications, including compound interest and population growth. Understanding how to manipulate and solve equations involving exponents is crucial for solving the given equation.

Logarithms

Logarithms are the inverse operations of exponentiation, defined as log_b(a) = c if and only if b^c = a. They help in solving equations where the variable is an exponent. In the context of the question, the logarithm base 2 is used, which simplifies the expression and allows for easier manipulation of the equation.

Properties of Logarithms

The properties of logarithms, such as the product, quotient, and power rules, are essential for simplifying logarithmic expressions. For instance, the power rule states that log_b(a^c) = c * log_b(a). These properties enable us to rewrite and solve logarithmic equations more efficiently, which is necessary for solving the equation presented.