/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 29 Evaluate the given quantities as... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 29

Evaluate the given quantities assuming that $$ \begin{array}{l} \log _{3} x=5.3 \text { and } \log _{3} y=2.1 \\ \log _{4} u=3.2 \text { and } \log _{4} v=1.3 \end{array} $$ $$ \log _{3} \frac{x^{3}}{y^{2}} $$

Short Answer

Expert verified
Using the logarithmic properties, the expression \(\log_3 \frac{x^3}{y^2}\) can be simplified to \(3 \log_3 x - 2 \log_3 y\). By substituting given values \(\log_3 x = 5.3\) and \(\log_3 y = 2.1\), the expression becomes \(15.9 - 4.2 = 11.7\). Therefore, \(\log_3 \frac{x^3}{y^2} = 11.7\).

Step by step solution

01

Recall the logarithmic properties

We will need the following logarithmic properties to simplify the expression: 1. \(\log_a (mn) = \log_a m + \log_a n\) 2. \(\log_a \frac{m}{n} = \log_a m - \log_a n\) 3. \(\log_a {m^n} = n \log_a m\)
02

Apply the exponent property of logarithms

Using the third property mentioned above, we will rewrite the given expression: \(\log_3 \frac{x^3}{y^2} = \log_3 (x^3) - \log_3 (y^2)\) Now we can apply the exponent property to each logarithm: \(3 \log_3 x - 2 \log_3 y\)
03

Substitute the given logarithmic values

Now we will substitute the given values of \(\log_3 x\) and \(\log_3 y\) into our simplified expression: \(3 \times 5.3 - 2 \times 2.1 = 15.9 - 4.2\)
04

Solve for the final expression value

Now, we will calculate the final value of the expression: \(15.9 - 4.2 = 11.7\) So, \(\log_3 \frac{x^3}{y^2} = 11.7\).

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Most popular questions from this chapter

At the end of \(2005,\) the largest known prime number was \(2^{30402457}-1 .\) How many digits does this prime number have?

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

Suppose you have a calculator that can only compute square roots and can multiply. Explain how you could use this calculator to compute \(7^{3 / 4}\).

Find the smallest integer \(M\) such that \(5^{1 / M}<1.01\).

Suppose you have a calculator that can only compute square roots. Explain how you could use this calculator to compute \(7^{1 / 8}\).
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