Problem 88 - A Graphical Approach to College Algebra

Question

Use the properties of logarithms to rewrite each expression as a single logarithm with coefficient 1. Assume that all variables represent positive real numbers. $$\frac{4}{3} \ln m-\frac{2}{3} \ln 8 n-\ln m^{3} n^{2}$$

Step-by-Step Solution

Answer

The single logarithm expression is: \( \ln \left( \frac{1}{4m^{\frac{5}{3}}n^{\frac{8}{3}}} \right) \).

Step 1: Distribute the Coefficients Into Logarithms

First, apply the property that allows you to move coefficients inside a logarithm. This property states that \( a \ln x = \ln x^a \). Applying this, we have: \( \ln m^{\frac{4}{3}} \) and \( \ln (8n)^{\frac{2}{3}} \).

Step 2: Rewrite the Expression Using Logarithm Properties

Using the expressions from Step 1, rewrite the initial equation: \( \ln m^{\frac{4}{3}} - \ln (8n)^{\frac{2}{3}} - \ln m^3n^2 \).

Step 3: Combine Using the Quotient Property of Logarithms

Use the property \( \ln a - \ln b = \ln \frac{a}{b} \). Let \( a = m^{\frac{4}{3}} \) and \( b = (8n)^{\frac{2}{3}} \), and further subtract \( \ln m^3n^2 \) to get:\[ \ln \left( \frac{m^{\frac{4}{3}}}{(8n)^{\frac{2}{3}}} \right) - \ln m^3n^2 \].

Step 4: Apply the Quotient Property Again

Combine the two logarithms from Step 3 using the quotient property again:\[ \ln \left( \frac{m^{\frac{4}{3}}}{(8n)^{\frac{2}{3}}} \times \frac{1}{m^3 n^2} \right) \].

Step 5: Simplify the Expression

Combine the powers of \( m \) and \( n \):The expression becomes:\[ \ln \left( \frac{m^{\frac{4}{3}}}{(8n)^{\frac{2}{3}}} \times \frac{1}{m^3 n^2} \right) = \ln \left( \frac{m^{\frac{4}{3} - 3}}{8^{\frac{2}{3}} n^{\frac{2}{3} + 2}} \right) \].Calculate the exponents:\( m^{\frac{4}{3}-3} = m^{-\frac{5}{3}} \) and \( 8^{\frac{2}{3}} = 4 \) (since \( 8 = 2^3 \), thus \( (2^3)^{\frac{2}{3}} = 2^2 = 4 \)).Combine and simplify:\[ \ln \left( \frac{m^{-\frac{5}{3}}}{4n^{\frac{8}{3}}} \right) = \ln \left( \frac{1}{4m^{\frac{5}{3}}n^{\frac{8}{3}}} \right) \].

Key Concepts

Coefficient Manipulation in Logarithms

When working with logarithms, one fundamental property is the ability to manipulate coefficients by incorporating them into the logarithmic argument as exponents. This is derived from the rule that states: \[ a \ln x = \ln x^a \]

• This property helps transform logarithmic expressions by removing coefficients.
• For instance, \( \frac{4}{3} \ln m \) becomes \( \ln m^{\frac{4}{3}} \).

This step is crucial for simplifying complex logarithmic expressions and sets the stage for applying other properties accurately. By understanding this manipulation, you gain better control over expressions, making them easier to work with and eventually simplify to a form with a single logarithm.

Logarithmic Quotient Property

The quotient property of logarithms is a key tool for combining or breaking apart logarithmic expressions. It states:\[ \ln a - \ln b = \ln \frac{a}{b} \]

• This property allows for the simplification of differences between logs into the log of a fraction.
• With the example, turning \( \ln a - \ln b \) into a more compact form \( \ln \frac{a}{b} \).

Applying this in the exercise allows us to simplify multi-term logarithmic expressions step-by-step. Like transforming:\[ \ln m^{\frac{4}{3}} - \ln (8n)^{\frac{2}{3}} \]into:\[ \ln \left( \frac{m^{\frac{4}{3}}}{(8n)^{\frac{2}{3}}}\right) \]Furthermore, you can apply the property again to handle additional terms, ensuring the expression moves progressively towards a single logarithm.

Simplifying Logarithmic Expressions

Simplifying logarithmic expressions involves combining all your knowledge of logarithm properties to re-write complex expressions in simpler forms. After using the properties of coefficients and the quotient property, you consolidate terms:

• Ensure each base and coefficient is properly combined and simplified.
• This includes performing calculations on exponents post-transformation.

Consider the logarithmic expression:\[ \ln \left( \frac{m^{\frac{4}{3} - 3}}{8^{\frac{2}{3}} n^{\frac{2}{3} + 2}} \right) \]You calculate:

• \( m^{\frac{4}{3} - 3} \rightarrow m^{\frac{-5}{3}} \)
• \( 8^{\frac{2}{3}} \rightarrow 4 \)

Combine and simplify to:\[ \ln \left( \frac{1}{4m^{\frac{5}{3}}n^{\frac{8}{3}}} \right) \] By following these steps, you condense a complex, multi-term logarithmic expression into a single, more manageable form, making it clearer and functional for further calculations.