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Chapter 10: Problem 257

In the following exercises, use the Properties of Logarithms to expand the logarithm. Simplify if possible. \(\log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}}\)

Short Answer

Expert verified
\frac{2}{3} \log_{3} x - 3 - 4 \log_{3} y.

Step by step solution

01

Identify the logarithm

The logarithm given is \( \log _{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} \). This fraction can be expanded using properties of logarithms.
02

Apply the quotient rule

Use the property \( \log_{b} \frac{M}{N} = \log_{b} M - \log_{b} N \). This gives us \( \log_{3} \sqrt[3]{x^{2}} - \log_{3} 27 y^{4} \).
03

Simplify the numerator

Rewrite \( \log_{3} \sqrt[3]{x^{2}} \) using \( \sqrt[3]{x^{2}} = x^{\frac{2}{3}} \). Therefore, \( \log_{3} \sqrt[3]{x^{2}} = \log_{3} x^{\frac{2}{3}} \).
04

Apply the power rule

Use the power property \( \log_{b} M^{p} = p \log_{b} M \). Thus, \( \log_{3} x^{\frac{2}{3}} = \frac{2}{3} \log_{3} x \).
05

Break down the denominator

Separate the log term \( \log_{3} 27 y^{4} \) using the product rule \( \log_{b} (MN) = \log_{b} M + \log_{b} N \), resulting in \( \log_{3} 27 + \log_{3} y^{4} \).
06

Simplify the denominator logs

Evaluate \( \log_{3} 27 = 3 \) because \( 27 = 3^{3} \), and apply the power rule to get \( \log_{3} y^{4} = 4 \log_{3} y \). Hence, \( \log_{3} 27 + \log_{3} y^{4} = 3 + 4 \log_{3} y \).
07

Combine all parts

Putting it all together, the expanded form is \( \frac{2}{3} \log_{3} x - (3 + 4 \log_{3} y) \).
08

Simplify the final expression

Distribute the negative sign: \( \frac{2}{3} \log_{3} x - 3 - 4 \log_{3} y \). This is the simplified expanded form.

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Key Concepts

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

Logarithm Expansion
Logarithm expansion is the process of breaking apart a complex logarithmic expression into simpler components. This can be done using various properties of logarithms.
For example, given a logarithmic expression involving a quotient or product, we can expand it by applying the appropriate rules. This makes it easier to simplify and understand.
In our exercise, the goal is to expand \( \log \_{3} \frac{\sqrt[3]{x^{2}}}{27 \ y^{4}} \). We use properties of logarithms to make the expression more manageable.
Quotient Rule in Logarithms
The Quotient Rule in logarithms allows you to separate the logarithm of a quotient into the difference of two logarithms. Mathematically, it is written as:
\( \log_{b} \frac{M}{N} = \log_{b} M - \log_{b} N \)
By applying this rule to our exercise, we can break down \( \log \_{3} \frac{\sqrt[3]{x^{2}}}{27 \ y^{4}} \) into separate components.
Therefore, \( \log_{3} \frac{\sqrt[3]{x^{2}}}{27 y^{4}} \) becomes \( \log_{3} \sqrt[3]{x^{2}} - \log_{3} 27 y^{4} \). This simplifies our work significantly.
Power Rule in Logarithms
The Power Rule in logarithms lets you move an exponent in the argument of a logarithm to in front of the logarithm. It is expressed as:
\( \log_{b} M^{p} = p \log_{b} M \)
Applying this rule helps to simplify expressions where the argument is raised to a power.
For example, \( \log_{3} x^{\frac{2}{3}} \) can be rewritten as \( \frac{2}{3} \log_{3} x \). This is very helpful in our exercise where we simplified \( \log_{3} \sqrt[3]{x^{2}} \).
Product Rule in Logarithms
The Product Rule in logarithms splits the logarithm of a product into the sum of the logarithms of its factors. It is given by:
\( \log_{b} (MN) = \log_{b} M + \log_{b} N \)
Using this rule, we can break down complex logarithmic expressions involving products.
In our exercise, we applied this rule to the denominator, \( 27 y^{4} \), in \( \log_{3} 27 y^{4} \). This resulted in \( \log_{3} 27 + \log_{3} y^{4} \), which we further simplified using the Power Rule and by evaluating \( \log_{3} 27 \).

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Most popular questions from this chapter

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(3 e^{x+2}=9\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(6 e^{2 x}=24\)

In the following exercises, solve each exponential equation. Find the exact answer and then approximate it to three decimal places. \(\frac{1}{3} e^{x}=2\)

In the following exercises, use the Quotient Property of Logarithms to write each logarithm as a sum of logarithms. Simplify if possible. \(\log \frac{x}{10}\)

In the following exercises, solve each equation. \(3 \log x=\log 125\)
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