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

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Recognize that the expression is of the form \(10^{\log 33}\), where the base of the logarithm is 10 (common logarithm).

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.

Apply this identity to the expression: since the base of the exponent and the base of the logarithm are both 10, \(10^{\log 33} = 33\).

Therefore, the expression simplifies directly to 33 without any further calculation.

This simplification works because the logarithm and the exponent are inverse operations when they share the same base.

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주요 개념

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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.

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