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Chapter 3: Problem 26

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{4} \frac{1}{\sqrt{v}} $$

Short Answer

Expert verified
After applying the properties of logarithms and substituting the given value, we find that \(\log_4 \frac{1}{\sqrt{v}} = -0.65\).

Step by step solution

01

Understand the properties of logarithms

When solving logarithm problems, it's important to be familiar with the basic properties of logarithms. Here are the properties we'll use in this problem: 1. \(\log_b(a^n) = n\log_b(a)\) 2. \(\log_b(\frac{a}{b}) = \log_b(a) - \log_b(b)\) 3. \(\log_b(\sqrt[n]{a}) = \frac{1}{n}\log_b(a)\)
02

Apply the properties to the expression

We need to evaluate \(\log_4 \frac{1}{\sqrt{v}}\). By applying property 2, we have: \[ \log_4\frac{1}{\sqrt{v}} = \log_4(1) - \log_4(\sqrt{v}) \] Since the logarithm of 1 in any base is always 0, the expression simplifies to: \[ \log_4\frac{1}{\sqrt{v}} = -\log_4(\sqrt{v}) \] Next, we apply property 3 to simplify the square root: \[ -\log_4(\sqrt{v}) = -\left(\frac{1}{2} \log_4({v}) \right) \]
03

Substitute the given value and compute the result

We are given that \(\log_4 v = 1.3\). Let's substitute this value in the expression we derived in the previous step: \[ -\left(\frac{1}{2} \log_4({v}) \right) = -\left(\frac{1}{2} \cdot 1.3 \right) \] Now calculate the result: \[ -\left(\frac{1}{2} \cdot 1.3 \right) = -0.65 \]
04

State the answer

After applying the properties of logarithms and substituting the given value, we find that: \[ \log_4 \frac{1}{\sqrt{v}} = -0.65 \]

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