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Ch. 7 - Transcendental Functions
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoHass 15th EditionCh. 7 - Transcendental FunctionsProblema 7.2.4c
Capitolo 7, Problema 7.2.4c

4. Use the properties of logarithms to write the expressions in Exercises 3 and 4 as a single term.
c. 3ln ∛(t² - 1) - ln(t+1)

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Identify the properties of logarithms that will help combine the terms: the power rule \(a \ln b = \ln b^a\) and the subtraction rule \(\ln A - \ln B = \ln \left( \frac{A}{B} \right)\).
Apply the power rule to the first term: rewrite \(3 \ln \sqrt[3]{t^2 - 1}\) as \(\ln \left( \sqrt[3]{t^2 - 1} \right)^3\).
Simplify the expression inside the logarithm by raising the cube root to the third power: \(\left( \sqrt[3]{t^2 - 1} \right)^3 = t^2 - 1\).
Rewrite the expression now as \(\ln (t^2 - 1) - \ln (t + 1)\).
Use the subtraction rule to combine the logarithms into a single term: \(\ln \left( \frac{t^2 - 1}{t + 1} \right)\).

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Properties of Logarithms

Logarithmic properties include rules such as the product, quotient, and power rules. These allow combining or breaking down logarithmic expressions, for example, ln(a) + ln(b) = ln(ab), ln(a) - ln(b) = ln(a/b), and k·ln(a) = ln(a^k). Understanding these is essential to rewrite expressions as a single logarithm.
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Change of Base Property

Exponent and Root Relationships

Roots can be expressed as fractional exponents, such as the cube root of x being x^(1/3). This allows rewriting expressions inside logarithms with roots into powers, facilitating the use of logarithm power rules to simplify or combine terms.
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Introduction to Exponent Rules

Combining Logarithmic Expressions

To write multiple logarithmic terms as a single term, apply the properties of logarithms step-by-step: first convert coefficients to exponents, then use product or quotient rules to combine terms. This process simplifies complex expressions into one logarithm.
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Logarithms Introduction
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