Solve each equation. $x=2^{{\log }_{2}9}$

Verified step by step guidance

1. Recognize that the expression involves an exponent with a logarithm: \(x = 2^{\log_2 9}\). The base of the exponent and the base of the logarithm are the same (both 2).
2. Recall the logarithmic identity: \(a^{\log_a b} = b\). This means that when the base of the exponent and the logarithm match, the expression simplifies directly to the argument of the logarithm.
3. Apply the identity to simplify \(2^{\log_2 9}\) to just 9.
4. Therefore, the solution to the equation is \(x = 9\).
5. Verify the solution by substituting back into the original expression to ensure consistency.

Key Concepts

Logarithmic and Exponential Functions

Logarithms and exponentials are inverse operations. The logarithm log_b(a) answers the question: to what power must the base b be raised to get a? Understanding this inverse relationship helps simplify expressions like 2^log2(9).

Properties of Logarithms

Key properties such as log_b(b^x) = x and b^{log_b(x)} = x allow simplification of expressions involving logs and exponents with the same base. These properties are essential to solve equations like x = 2^{log2(9)}.

Evaluating Expressions with Same Base

When the base of the exponent and the base of the logarithm are the same, the expression simplifies directly to the argument of the logarithm. For example, 2^{log2(9)} simplifies to 9, which is crucial for solving the given equation.