Evaluate or simplify each expression without using a calculator. 10^{log 33}

Verified step by step guidance

1. Recognize that the expression is of the form \(10^{\log 33}\), where the base of the logarithm is 10 (common logarithm).
2. Recall the logarithmic identity: \(a^{\log_a x} = x\), which means that raising a base to the logarithm of a number with the same base returns the number itself.
3. Apply this identity to the expression: since the base of the exponent and the base of the logarithm are both 10, \(10^{\log 33} = 33\).
4. Therefore, the expression simplifies directly to 33 without any further calculation.
5. This simplification works because the logarithm and the exponent are inverse operations when they share the same base.

Key Concepts

Logarithmic and Exponential Functions

Logarithmic and exponential functions are inverse operations. The logarithm log_b(x) answers the question: to what power must the base b be raised to get x? Understanding this inverse relationship is key to simplifying expressions involving logs and exponents.

Properties of Logarithms and Exponents

Key properties include that b^(log_b(x)) = x, meaning raising a base to the logarithm of a number with the same base returns the number itself. This property allows simplification of expressions like 10^(log 33) directly to 33 without calculation.

Evaluating Expressions Without a Calculator

Simplifying expressions without a calculator often relies on recognizing patterns and applying algebraic properties. For 10^(log 33), knowing the base and log relationship avoids numeric approximation, enabling exact simplification.