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Chapter 5: Problem 13

Write the expression as one logarithm. $$ \log \left(x^{3} y^{2}\right)-2 \log x \sqrt[3]{y}-3 \log \left(\frac{x}{y}\right) $$

Short Answer

Expert verified
\( \log \left( \frac{y^{13/3}}{x^2} \right) \)

Step by step solution

01

Apply the Power Rule of Logarithms

Recall that the power rule of logarithms states that \( a \log b = \log b^a \). Apply this rule to each term in the expression:\[2\log x \sqrt[3]{y} = \log (x\sqrt[3]{y})^2 = \log (x^2 y^{2/3}) \]\[3 \log \left(\frac{x}{y}\right) = \log \left(\frac{x}{y}\right)^3 = \log \left(\frac{x^3}{y^3}\right) \]The expression now becomes:\[\log (x^3 y^2) - \log (x^2 y^{2/3}) - \log \left(\frac{x^3}{y^3}\right)\]
02

Apply the Quotient Rule of Logarithms

The quotient rule of logarithms states that \( \log a - \log b = \log \frac{a}{b} \). Apply this rule to the modified expression:\[ \log \frac{x^3 y^2}{x^2 y^{2/3}} - \log \left(\frac{x^3}{y^3}\right) \]
03

Simplify the Expression

Simplify the expression within each logarithm by performing division and applying the laws of exponents:\[ \log \frac{x^3 y^2}{x^2 y^{2/3}} = \log \left( x^{3-2} y^{2-(2/3)} \right) = \log \left( x^1 y^{4/3} \right) \]Now substitute back into the expression:\[ \log (x y^{4/3}) - \log \left(\frac{x^3}{y^3}\right) \]
04

Combine the Logarithms Using the Quotient Rule Again

Combine the two logarithms using the quotient rule once more:\[ \log \frac{x y^{4/3}}{\frac{x^3}{y^3}} \]Perform division:\[ \frac{x y^{4/3} \cdot y^3}{x^3} = \frac{y^{4/3 + 3}}{x^{3-1}} = \frac{y^{13/3}}{x^2} \]
05

Final Step: Express as One Logarithm

The final expression simplifies to:\[ \log \left( \frac{y^{13/3}}{x^2} \right) \]

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Key Concepts

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

Power Rule of Logarithms
The Power Rule of Logarithms is a fundamental tool in simplifying complex logarithmic expressions. This rule states that if you have an expression of the form \( a \log b \), you can rewrite it as \( \log b^a \). This is especially useful when dealing with logarithms involving exponents.

Imagine you have \( 2\log x \sqrt[3]{y} \). Using the power rule, this can be changed into \( \log (x \sqrt[3]{y})^2 \). This becomes \( \log (x^2 y^{2/3}) \) by applying the power inside the logarithm to both \( x \) and \( y \).
  • It simplifies expressions by combining multiplication and exponentiation under a single logarithm.
  • Keeps calculations tidy by reducing the number of operations you have to perform.
With this rule, you're able to tame otherwise unwieldy logarithmic terms by transforming them into a more manageable format.
Quotient Rule of Logarithms
The Quotient Rule of Logarithms is a powerful mechanism for handling subtraction between logarithmic expressions. It is expressed by the formula \( \log a - \log b = \log \frac{a}{b} \). This allows a subtraction of logs to be combined into a single log.

In practice, consider the expression \( \log (x^3 y^2) - \log (x^2 y^{2/3}) \). We can apply the quotient rule to write it as \( \log \frac{x^3 y^2}{x^2 y^{2/3}} \).
  • Useful for simplifying expressions, making them more compact.
  • Transforms complex subtractive relationships in logs into division inside a single logarithm.
Utilizing this rule ensures that you can navigate logarithmic expressions with multiple components efficiently.
Laws of Exponents
The Laws of Exponents play a crucial role when handling expressions inside a logarithm, especially during simplification. These laws govern how we combine and manipulate powers of the same base.

In the context of logarithms, let's say you have \( \frac{x^3 y^2}{x^2 y^{2/3}} \). Here, you need to apply the laws such as:
  • When dividing like bases, you subtract their exponents: \( x^{3-2} = x^1 \).
  • When dealing with \( y \), \( y^{2-(2/3)} = y^{4/3} \).
By using these laws, the base elements inside the fractional expression are simplified, resulting in fewer terms which makes further operations and understanding more straightforward.
Simplifying Logarithms
Simplifying logarithmic expressions involves applying rules systematically until you reach the simplest form possible. This means combining rules of logarithms and exponents to condense everything into a single, clear expression.

Take for instance transforming \( \log \frac{x y^{4/3}}{\frac{x^3}{y^3}} \) into a simplified form. By performing careful division, you manage to reorganize it to \( \frac{y^{13/3}}{x^2} \).
  • Process includes employing exponent and log rules strategically.
  • Ensures maximum reduction of expression complexity making further mathematical manipulation easier.
The simplification process always aims to express your result in the most compact form, minimizing cognitive load when analyzing or solving problems.

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Most popular questions from this chapter

The energy \(E(x)\) of an electron after passing through material of thickness \(x\) is given by the equation \(E(x)=E_{0} e^{-x / x_{0}}\), where \(E_{0}\) is the initial energy and \(x_{0}\) is the radiation length. (a) Express, in terms of \(E_{0}\), the energy of an electron after it passes through material of thickness \(x_{0}\). (b) Express, in terms of \(x_{0}\), the thickness at which the electron loses \(99 \%\) of its initial energy.

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Pareto's law for capitalist countries states that the relationship between annual income \(x\) and the number \(y\) of individuals whose income exceeds \(x\) is $$ \log y=\log b-k \log x, $$ where \(b\) and \(k\) are positive constants. Solve this equation for \(y\).
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