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Chapter 1: Problem 22

Write the given quantity in terms of \(\log x, \log y,\) and \(\log z\). \(\log \frac{x^{2} y^{3}}{\sqrt{z}}\)

Short Answer

Expert verified
\(2\log x + 3\log y - \frac{1}{2}\log z\)

Step by step solution

01

Apply the Logarithm of a Quotient Rule

Start by applying the rule \(\log \frac{a}{b} = \log a - \log b\). Here, we have \(a = x^2 y^3\) and \(b = \sqrt{z}\). Thus, the expression becomes: \[ \log \frac{x^{2} y^{3}}{\sqrt{z}} = \log(x^2 y^3) - \log(\sqrt{z})\]
02

Apply the Logarithm of a Product Rule

Now, apply the product rule \(\log (a \cdot b) = \log a + \log b\) to the terms inside \(\log(x^2 y^3)\). This gives us:\[ \log (x^2 y^3) = \log x^2 + \log y^3\]
03

Apply the Power Rule of Logarithms

Use the power rule \(\log a^b = b \log a\) for each term. Start with \(\log x^2\) and \(\log y^3\):\[\log x^2 = 2\log x \]\[\log y^3 = 3\log y\]
04

Simplify the Numerator Logarithm

Combine the results from the previous step:\[\log (x^2 y^3) = 2\log x + 3\log y\]
05

Simplify the Denominator Logarithm

Apply the power rule to \(\log(\sqrt{z})\) using \(\sqrt{z} = z^{1/2}\):\[\log(\sqrt{z}) = \frac{1}{2}\log z\]
06

Assemble the Final Expression

Subtract the logarithm of the denominator from the numerator:\[\log \frac{x^{2} y^{3}}{\sqrt{z}} = (2\log x + 3\log y) - \frac{1}{2}\log z\]Simplified, this becomes:\[2\log x + 3\log y - \frac{1}{2}\log z\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Logarithm Quotient Rule
When dealing with logarithms, one useful rule is the logarithm quotient rule. This rule states that the logarithm of a quotient, such as \( \frac{a}{b} \), can be expressed as the difference of two logarithms: \( \log \frac{a}{b} = \log a - \log b \). In other words, to find the logarithm of a fraction, simply subtract the logarithm of the denominator from the logarithm of the numerator.

For example, if you have a logarithmic expression like \( \log \frac{x^2 y^3}{\sqrt{z}} \), you can apply the quotient rule to break it down:
  • The expression becomes \( \log(x^2 y^3) - \log(\sqrt{z}) \).
  • This simplifies your work by separating the expression into two manageable parts.
The beauty of the quotient rule is that it makes seemingly complex expressions easier to handle by straightforward subtraction.
Logarithm Product Rule
The logarithm product rule is another essential tool in simplifying logarithmic expressions. It helps us express the logarithm of a product of numbers as the sum of the logarithms of those numbers. The rule can be stated as: \( \log (ab) = \log a + \log b \).

This rule comes in handy when you have a term like \( \log (x^2 y^3) \). You can apply the product rule to break it into smaller parts:
  • Start with \( a = x^2 \) and \( b = y^3 \), leading to \( \log(x^2 y^3) = \log x^2 + \log y^3 \).
  • Now, instead of a single complex term, you have two simpler logarithms to work with.
This approach helps transform a product into a sum, which often makes further simplification more convenient.
Logarithm Power Rule
The logarithm power rule is a straightforward yet powerful concept. It allows you to simplify logarithmic expressions where a number is raised to a power. According to this rule, \( \log a^b = b \log a \). Essentially, you bring the exponent in front of the logarithm as a multiplier.

Consider an expression like \( \log x^2 + \log y^3 \). Applying the power rule:
  • Change \( \log x^2 \) to \( 2\log x \).
  • Convert \( \log y^3 \) to \( 3\log y \).
Similarly, you can handle the term \( \log(\sqrt{z}) \) using the power representation \( \sqrt{z} = z^{1/2} \). Thus, it becomes:
  • \( \log(\sqrt{z}) = \frac{1}{2}\log z \).
With the power rule, powers become simple multipliers, greatly simplifying the original expressions.

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Most popular questions from this chapter

Featherstone and coauthors \(^{37}\) studied 195 Kansas beef cow farms. The average fixed and variable costs are found in the following table. $$ \begin{array}{|c|r|} \hline {\text { Variable and Fixed Costs }} \\ \hline \text { Costs per cow } & \\ \text { Feed costs } & \$ 261 \\ \text { Labor costs } & \$ 82 \\ \text { Utilities and fuel costs } & \$ 19 \\ \text { Veterinary expenses costs } & \$ 13 \\ \text { Miscellaneous costs } & \$ 18 \\ \hline \text { Total variable costs } & \$ 393 \\ \text { Total fixed costs } & \$ 13,386 \\ \hline \end{array} $$ The farm can sell each cow for \(\$ 470 .\) Find the cost, revenue, and profit functions for an average farm. The average farm had 97 cows. What was the profit for 97 cows? Can you give a possible explanation for your answer?

Hardman \(^{51}\) showed the survival rate \(S\) of European red mite eggs in an apple orchard after insect predation was approximated by $$ y=\left\\{\begin{array}{ll} 1, & t \leq 0 \\ 1-0.01 t-0.001 t^{2}, & t>0 \end{array}\right. $$ where \(t\) is the number of days after June \(1 .\) Determine the predation rate on May \(15 .\) On June \(15 .\)

You are given a pair of functions, \(f\) and \(g .\) In each case, use your grapher to estimate the domain of \((g \circ f)(x)\). Confirm analytically. $$ f(x)=x^{2}, g(x)=\sqrt{x} $$

Your rich uncle has just given you a high school graduation present of \(\$ 1\) million. The present, however, is in the form of a 40 -year bond with an annual interest rate of \(9 \%\) compounded annually. The bond says that it will be worth \(\$ 1\) million in 40 years. What is this million-dollar gift worth at the present time?

Find the effective yield given the annual rate \(r\) and the indicated compounding. \(r=10 \%,\) compounded (a) semiannually, (b) quarterly, (c) monthly, (d) weekly, (e) daily, (f) continuously.
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