Evaluate or simplify each expression without using a calculator. log 10⁷

검증된 단계별 안내

Recall the definition of logarithms: $\log_b(a)$ asks the question, "To what power must the base $b$ be raised to get $a$?"

In this problem, the base of the logarithm is 10, and the argument is \$10^7$, so we are looking for the exponent $x$ such that $10^x = 10^7$.

Since the bases on both sides of the equation are the same (base 10), the exponents must be equal. Therefore, $x = 7$.

Thus, $\log 10^{7} = 7$ because the logarithm of a power of 10 with base 10 is simply the exponent.

This simplification uses the logarithmic identity: $\log_b(b^k) = k$ for any positive base $b \neq 1$ and any real number $k$.

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주요 개념

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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_b(a) = c means b^c = a. Understanding this definition is essential for evaluating logarithmic expressions.

Logarithm of a Power

The logarithm of a number raised to an exponent can be simplified using the rule log_b(a^n) = n * log_b(a). This property allows you to bring the exponent in front as a multiplier, simplifying calculations without a calculator.

Common Logarithms (Base 10)

Common logarithms have base 10, written as log or log_10. Since log_10(10) = 1, powers of 10 simplify easily: log_10(10^n) = n. Recognizing this helps quickly evaluate expressions like log 10^7.

Evaluate or simplify each expression without using a calculator. log 107