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Chapter 5: Problem 27

Express in terms of sums and differences of logarithms. $$\ln \frac{2}{3 x^{3} y}$$

Short Answer

Expert verified
\( ln \frac{2}{3 x^{3} y} = \ln 2 - \ln 3 - 3 \ln x - \ln y\)

Step by step solution

01

Understand the Problem

You need to express \(\ln \frac{2}{3 x^{3} y}\) in terms of sums and differences of logarithms. To do this, use the properties of logarithms such as the quotient rule and the product rule.
02

Apply the Quotient Rule

The quotient rule states that \(\ln \frac{a}{b} = \ln a - \ln b\). Here, apply this rule to split the logarithm into two parts: \(\ln \frac{2}{3 x^{3} y} = \ln 2 - \ln (3 x^{3} y)\).
03

Apply the Product Rule

The product rule states that \(\ln (abc) = \ln a + \ln b + \ln c\). Apply this rule to \(\ln (3 x^{3} y)\): \(\ln (3 x^{3} y) = \ln 3 + \ln x^{3} + \ln y\).
04

Apply the Power Rule

The power rule states that \( ln (x^{n}) = n \ln x\). Apply this rule to \( ln x^{3}\): \( ln x^{3} = 3\ln x\).
05

Combine All Parts

Now substitute back the logarithms: \( ln \frac{2}{3 x^{3} y} = \ln 2 - (\ln 3 + 3 \ln x + \ln y) \). Simplify by distributing the negative sign: \( ln \frac{2}{3 x^{3} y} = \ln 2 -\ln 3 - 3\ln x -\ln y\).

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Key Concepts

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

Quotient Rule in Logarithms
The quotient rule is extremely useful when you need to simplify the logarithm of a fraction. This rule states that the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator. In mathematical terms, it is expressed as:
\[ \ln \frac{a}{b} = \ln a - \ln b \]
When applying this rule to our original problem, \(\ln \frac{2}{3 x^{3} y}\), we divide it into:
\[ \ln \frac{2}{3 x^{3} y} = \ln 2 - \ln (3 x^{3} y) \]
This splitting helps us break down the problem into simpler parts that are easier to manage.
Product Rule in Logarithms
The product rule is another essential property of logarithms. This rule states that the logarithm of a product is the sum of the logarithms of the individual factors. Mathematically, it can be expressed as:
\[ \ln (abc) = \ln a + \ln b + \ln c \]
Applying the product rule to our intermediate expression \(\ln (3 x^{3} y)\) yields:
\[ \ln (3 x^{3} y) = \ln 3 + \ln x^{3} + \ln y \]
By using the product rule, we can further break down the logarithms into simpler terms that can be handled individually.
Power Rule in Logarithms
Finally, the power rule is crucial when dealing with logarithms of terms raised to an exponent. The rule states that the logarithm of a power is the exponent times the logarithm of the base. The mathematical representation is:
\[ \ln (x^{n}) = n \ln x \]
We apply the power rule to the term \(\ln x^{3}\) from our intermediate expression:
\[ \ln x^{3} = 3 \ln x \]
With the power rule, we simplify logarithmic expressions where variables are raised to a power, making them easier to manage.

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Most popular questions from this chapter

Interest in a College Trust Fund. Following the birth of his child, Benjamin deposits \(\$ 10,000\) in a college trust fund where interest is \(3.9 \%,\) compounded semiannually. a) Find a function for the amount in the account after \(t\) years. b) Find the amount of money in the account at \(t=0,4\) \(8,10,18,\) and 21 years.

Using only a graphing calculator, determine whether the functions are inverses of each other. $$f(x)=\frac{2 x-5}{4 x+7}, g(x)=\frac{7 x-4}{5 x+2}$$

Solve using any method. $$x\left(\ln \frac{1}{6}\right)=\ln 6$$

Solve. $$\log _{6} x=1-\log _{6}(x-5)$$

Alfalfa Imported by China. The amount and quality of milk produced increases when dairy cows are fed high-quality alfalfa. Although annual milk consumption per capita in China has been one of the lowest in the world, it has increased greatly in recent years. Consequently, the demand for imported alfalfa has increased exponentially. In \(2008,\) approximately \(20,000\) tons of U.S. alfalfa were imported by China. This number increased to approximately \(650,000\) tons by \(2013 .\) (Sources: National Geographic\(,\) May \(2015,\) Arjen Hoekstra, University of Twente; USDA Economic Research Service; Shefali Sharma and Zhang Rou, Institute for Agriculture and Trade Policy; FAO; Ministry of Agriculture, People's Republic of China) The increase in imported U.S. alfalfa can be modeled by the exponential function $$ A(x)=22,611.008(1.992)^{x} $$ where \(x\) is the number of years after 2008 . Find the number of tons of U.S. alfalfa imported by China in 2012 and in 2016 (IMAGE CANT COPY)
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