Evaluate or simplify each expression without using a calculator. log 100

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Recall the definition of the logarithm: \(\log_b a = c\) means that \(b^c = a\). Here, the base is 10 since it is a common logarithm (log without a base is base 10).

Rewrite the expression \(\log 100\) as \(\log_{10} 100\) to emphasize the base 10.

Express 100 as a power of 10: \(100 = 10^2\).

Use the logarithm power rule: \(\log_b (a^c) = c \log_b a\). Applying this, \(\log_{10} (10^2) = 2 \log_{10} 10\).

Since \(\log_{10} 10 = 1\), simplify the expression to \(2 \times 1 = 2\).

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Definition of Logarithms

A logarithm answers the question: to what exponent must the base be raised to produce a given number? For example, log base 10 of 100 asks, '10 raised to what power equals 100?' Understanding this definition is fundamental to evaluating logarithmic expressions.

Common Logarithms (Base 10)

Common logarithms use base 10 and are often written simply as log without a base. Since 100 = 10^2, log 100 means log base 10 of 100, which equals 2. Recognizing powers of 10 helps simplify these expressions quickly.

Properties of Logarithms

Logarithms have properties like log(a^b) = b * log(a), which allow simplification of expressions. Applying these properties helps evaluate or simplify logarithmic expressions without a calculator by rewriting numbers as powers of the base.