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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 37
Capitolo 5, Problema 37

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. ln(x3x2+1(x+1)4)\(\ln\) \(\left\)( \(\frac{x^3 \sqrt{x^2 + 1}\)}{(x + 1)^4} \(\right\))

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Start by writing the given logarithmic expression clearly: \(\ln\left( \frac{x^{3} \sqrt{x^{2} + 1}}{(x + 1)^{4}} \right)\).
Use the logarithm property for quotients: \(\ln\left( \frac{A}{B} \right) = \ln(A) - \ln(B)\), to separate the expression into two logarithms: \(\ln\left(x^{3} \sqrt{x^{2} + 1}\right) - \ln\left((x + 1)^{4}\right)\).
Next, apply the logarithm property for products: \(\ln(AB) = \ln(A) + \ln(B)\), to expand \(\ln\left(x^{3} \sqrt{x^{2} + 1}\right)\) into \(\ln(x^{3}) + \ln\left(\sqrt{x^{2} + 1}\right)\).
Rewrite the square root as an exponent: \(\sqrt{x^{2} + 1} = (x^{2} + 1)^{1/2}\), and use the power rule for logarithms: \(\ln\left(A^{r}\right) = r \ln(A)\), to express \(\ln\left(\sqrt{x^{2} + 1}\right)\) as \(\frac{1}{2} \ln(x^{2} + 1)\).
Similarly, apply the power rule to \(\ln(x^{3})\) and \(\ln\left((x + 1)^{4}\right)\) to get \(3 \ln(x)\) and \(4 \ln(x + 1)\) respectively. Combine all parts to write the fully expanded expression.

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Properties of Logarithms

Properties of logarithms include the product, quotient, and power rules, which allow the expansion or simplification of logarithmic expressions. For example, ln(ab) = ln(a) + ln(b), ln(a/b) = ln(a) - ln(b), and ln(a^n) = n ln(a). These rules help break down complex expressions into simpler sums and differences of logarithms.
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Simplifying Radicals and Exponents

Simplifying expressions involving radicals and exponents is essential before applying logarithmic properties. For instance, the square root √(x² + 1) can be left as is, but recognizing powers like x³ or (x + 1)^4 helps in applying the power rule of logarithms effectively.
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Domain Considerations for Logarithmic Functions

The domain of a logarithmic function requires the argument to be positive. When expanding ln[(x³(√(x² + 1)))/(x + 1)⁴], it is important to consider values of x that keep the entire expression inside the logarithm positive to ensure the expression is defined.
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