Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $\log_{b} (\frac{\sqrt{x} y^{3}}{z^{3}})$

검증된 단계별 안내

Start with the given logarithmic expression: $\log_{b} \left( \frac{\sqrt{x} y^{3}}{z^{3}} \right)$.

Use the logarithm property for division: $\log_{b} \left( \frac{M}{N} \right) = \log_{b} M - \log_{b} N$. So rewrite the expression as $\log_{b} (\sqrt{x} y^{3}) - \log_{b} (z^{3})$.

Apply the logarithm property for multiplication: $\log_{b} (MN) = \log_{b} M + \log_{b} N$. So expand $\log_{b} (\sqrt{x} y^{3})$ as $\log_{b} (\sqrt{x}) + \log_{b} (y^{3})$.

Rewrite the radicals and exponents inside the logarithms using the power rule: $\log_{b} (x^{1/2}) + \log_{b} (y^{3}) - \log_{b} (z^{3})$.

Use the power rule of logarithms: $\log_{b} (a^{c}) = c \log_{b} (a)$, to write the expression as $\frac{1}{2} \log_{b} (x) + 3 \log_{b} (y) - 3 \log_{b} (z)$.

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주요 개념

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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, log_b(MN) = log_b(M) + log_b(N), log_b(M/N) = log_b(M) - log_b(N), and log_b(M^k) = k·log_b(M). These rules help break down complex expressions into simpler parts.

Radicals and Exponents in Logarithms

Radicals can be rewritten as fractional exponents, such as √x = x^(1/2). This conversion allows the use of the power rule of logarithms to simplify expressions involving roots. Recognizing and converting radicals is essential for expanding logarithmic expressions fully.

Simplifying Logarithmic Expressions Without a Calculator

When possible, logarithmic expressions should be simplified using algebraic manipulation rather than numerical approximation. This involves applying logarithm properties to rewrite expressions in terms of simpler logs or constants, enabling exact answers and deeper understanding of the expression's structure.

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. logb(xy3z3)\(\log\)_{b}\(\left\)(\(\frac{\sqrt{x}\)y^3}{z^3}\(\right\))