Problem 22 Evaluate the given quantities as... [FREE SOLUTION]

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Precalculus A Prelude to Calculus

Sheldon Axler

$Math Studyset 91影视 Explanations$ Math

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Chapter 3: Problem 22

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{4} \frac{u}{8 v} $$

Short Answer

Expert verified

The short answer is: $\log _{4} \frac{u}{8 v} = 0.4$.

Step by step solution

Analyze the given logarithms

We are given the following logarithm values: $\\ \log _{3} x = 5.3 \Rightarrow x = 3^{5.3} \\ \log_{3} y = 2.1 \Rightarrow y = 3^{2.1} \\ \log _{4} u = 3.2 \Rightarrow u = 4^{3.2} \\ \log _{4} v = 1.3 \Rightarrow v = 4^{1.3} $

Write down the given expression

We need to evaluate: $\\ \log _{4} \frac{u}{8 v} $

Replace $u$ and $v$ with their expressions in terms of logarithms

Substitute the expressions for $u$ and $v$ from Step 1: $\\ \log _{4} \frac{4^{3.2}}{8 \cdot 4^{1.3}} $

Simplify the expression

Use logarithm properties and simplify: $\\ \log _{4} \frac{4^{3.2}}{4^{\log _{4} 8} \cdot 4^{1.3}} \Rightarrow \log _{4} \frac{4^{3.2}}{4^{(1.3 + \log _{4} 8)}} $

Use log properties to combine exponents

Use the property $\log _{a} b - \log _{a} c = \log _{a}(\frac{b}{c})$ to combine exponents: $\\ \log _{4} 4^{(3.2 - 1.3 - \log _{4} 8)} $

Evaluate the logarithm

Evaluate the logarithm: $\\ 3.2 - 1.3 - \log _{4} 8 = 3.2 - 1.3 - 1.5 = 0.4 $ Hence, the expression $\log _{4} \frac{u}{8 v} = 0.4$.

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91影视

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{4} \frac{u}{8 v} $$

Short Answer

Step by step solution

Analyze the given logarithms

Write down the given expression

Replace \(u\) and \(v\) with their expressions in terms of logarithms

Simplify the expression

Use log properties to combine exponents

Evaluate the logarithm

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