Solve each equation. $x=3{\log }_{3}8$

Verified step by step guidance

1. Recognize that the equation is given as \(x = 3^{\log_3 8}\), where the base of the exponent and the base of the logarithm are the same (both 3).
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 this identity to simplify \(3^{\log_3 8}\) to just 8.
4. Therefore, the value of \(x\) is equal to 8.
5. Conclude that the solution to the equation is \(x = 8\).

Key Concepts

Logarithmic and Exponential Functions

Logarithmic functions are the inverses of exponential functions. Understanding how these two relate helps in simplifying expressions like 3^log3(8), where the base of the exponent and the logarithm are the same.

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 for solving equations like x = 3^{log_3(8)}.

Evaluating Expressions with Matching Bases

When the base of the exponent matches the base of the logarithm, the expression simplifies directly to the argument of the logarithm. For example, 3^{log_3(8)} simplifies to 8, which is crucial for solving the given equation.