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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 23
Capitolo 5, Problema 23

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. log4(x64)\(\log\)_4\(\left\)(\(\frac{\sqrt{x}\)}{64}\(\right\))

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Recall the logarithmic property for quotients: \(\log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N\). Apply this to rewrite \(\log_4 \left( \frac{\sqrt{x}}{64} \right)\) as \(\log_4 (\sqrt{x}) - \log_4 (64)\).
Express the square root as an exponent: \(\sqrt{x} = x^{\frac{1}{2}}\). Use the power rule of logarithms: \(\log_b (M^p) = p \log_b M\). So, \(\log_4 (\sqrt{x}) = \log_4 (x^{\frac{1}{2}}) = \frac{1}{2} \log_4 x\).
Rewrite 64 as a power of 4 if possible. Since \(4^3 = 64\), we have \(64 = 4^3\). Use the power rule again: \(\log_4 (64) = \log_4 (4^3) = 3 \log_4 4\).
Recall that \(\log_b b = 1\) for any base \(b\). Therefore, \(\log_4 4 = 1\), so \(\log_4 (64) = 3 \times 1 = 3\).
Combine all parts to write the expanded form: \(\log_4 \left( \frac{\sqrt{x}}{64} \right) = \frac{1}{2} \log_4 x - 3\).

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Properties of Logarithms

Properties of logarithms include rules such as the product, quotient, and power rules. These allow us to rewrite logarithmic expressions by expanding or condensing them. For example, log_b(M/N) = log_b(M) - log_b(N) and log_b(M^k) = k * log_b(M). These properties are essential for simplifying and expanding logarithmic expressions.
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Change of Base Property

Radicals and Exponents

Understanding how to express radicals as fractional exponents is crucial. For instance, the square root of x can be written as x^(1/2). This conversion allows the use of logarithm power rules to simplify expressions involving roots. Recognizing this helps in expanding logarithmic expressions involving radicals.
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Rational Exponents

Change of Base and Evaluating Logarithms

Evaluating logarithms without a calculator often involves expressing numbers as powers of the base. For example, 64 can be written as 4^3 since 4^3 = 64. This allows direct simplification of logarithmic terms like log_4(64) = 3. This concept helps in simplifying and evaluating logarithmic expressions exactly.
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Change of Base Property
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