Domanda del libro di testo
In Exercises 105–108, evaluate each expression without using a calculator. log5 (log2 32)
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Ch. 4 - Exponential and Logarithmic Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 5, Problema 108
In Exercises 105–108, evaluate each expression without using a calculator. log (ln e)
Guida verificata passo dopo passo1
Recognize the expression: \( \log(\ln e) \). Here, \( \ln e \) means the natural logarithm of \( e \), and \( \log \) typically refers to the logarithm base 10.
Evaluate the inner natural logarithm first: \( \ln e \). Recall that \( \ln e = 1 \) because the natural logarithm of \( e \) (Euler's number) is 1.
Substitute the value back into the expression: \( \log(1) \). Now the problem simplifies to finding \( \log(1) \).
Recall the property of logarithms: for any base \( b \), \( \log_b(1) = 0 \) because \( b^0 = 1 \).
Therefore, \( \log(1) = 0 \). So the value of the original expression \( \log(\ln e) \) is 0.
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Natural Logarithm (ln)
The natural logarithm, denoted as ln, is the logarithm to the base e, where e ≈ 2.718. It answers the question: to what power must e be raised to get a certain number? For example, ln(e) = 1 because e¹ = e.
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2:51
The Natural Log
Logarithm Properties
Logarithms have properties that simplify expressions, such as log(a^b) = b log(a) and log(1) = 0. Understanding these properties helps evaluate nested logarithmic expressions without a calculator.
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5:36Change of Base Property
Relationship Between log and ln
The notation log often implies base 10, while ln is base e. Evaluating expressions like log(ln e) requires first finding ln e, then applying the base-10 logarithm to that result.
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2:51The Natural Log
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