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Ch. 4 - Exponential and Logarithmic Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 4 - Exponential and Logarithmic FunctionsProblema 83
Capitolo 5, Problema 83

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

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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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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.
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Logarithms Introduction

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.
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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.
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Graphs of Common Functions
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