Retaining the Concepts. Expand: log⁡7(5x49y10)\$$log_7$$ \left( \frac{5\sqrt{x}}{$$49y^{10}$$} \right)

Ch. 8 - Sequences, Induction, and Probability

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Blitzer 8th Edition

Ch. 8 - Sequences, Induction, and Probability

문제 97

9장, 문제 97

Retaining the Concepts. Expand: $\log_{7} (\frac{5 \sqrt{x}}{49 y^{10}})$

검증된 단계별 안내

Rewrite the expression clearly: you have $ \log_7 \left( \frac{5 \sqrt{x}}{49 y^{10}} \right) $ multiplied by the fifth root of $ x $, which can be written as $ x^{\frac{1}{5}} $.

Express the square root and fifth root in exponential form: $ \sqrt{x} = x^{\frac{1}{2}} $ and $ x^{\frac{1}{5}} $ remains as is.

Rewrite the argument inside the logarithm using exponents: $ \frac{5 x^{\frac{1}{2}}}{49 y^{10}} $.

Use the logarithm property for division: $ \log_7 \left( \frac{A}{B} \right) = \log_7 A - \log_7 B $. So, split the log into $ \log_7 (5 x^{\frac{1}{2}}) - \log_7 (49 y^{10}) $.

Apply the logarithm product and power rules: $ \log_7 (5) + \log_7 (x^{\frac{1}{2}}) - \left( \log_7 (49) + \log_7 (y^{10}) \right) $. Then, use the power rule $ \log_b (a^c) = c \log_b (a) $ to bring down exponents.

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주요 개념

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Properties of Logarithms

Logarithmic properties such as the product, quotient, and power rules allow us to expand or simplify logarithmic expressions. For example, log_b(MN) = log_b(M) + log_b(N), and log_b(M^k) = k·log_b(M). These rules help break down complex expressions into sums and differences of simpler logs.

Radicals and Exponents

Radicals like roots can be rewritten as fractional exponents, e.g., the fifth root of x is x^(1/5). Understanding this conversion is essential for applying logarithm power rules and simplifying expressions involving roots within logarithms.

Simplifying Algebraic Expressions Inside Logarithms

Before expanding, it is important to simplify the expression inside the logarithm by factoring and rewriting terms. This includes expressing constants as powers of the base if possible and separating products and quotients to apply logarithm rules effectively.

Retaining the Concepts. Expand: log7(5x49y10)\(\log\)_7 \(\left\)( \(\frac{5\sqrt{x}\)}{49y^{10}} \(\right\))