Problem 62 - College Algebra Essentials

Question

Use properties of logarithms to condense logarithmic expression. Write the expression as a single logarithm whose coefficient is 1. Where possible, evaluate logarithmic expressions without using a calculator. \(4 \ln x+7 \ln y-3 \ln z\)

Step-by-Step Solution

Answer

The condensed form of the logarithmic expression is \(\ln ((x^4y^7) / z^3)\)

Step 1: Apply the property of logarithm of a power

First, apply the property \(\(a \ln b = \ln (b^a)\) to each term in the expression so that it can be rewritten as \(\ln (x^4) + \ln (y^7) - \ln (z^3)\)

Step 2: Apply the property of logarithm of a product

Next, apply the property \(\ln a + \ln b = \ln (a \cdot b)\) to the terms that are being added to get \(\ln (x^4y^7) - \ln (z^3)\)

Step 3: Apply the property of logarithm of a division

Now, apply the property \(\ln a - \ln b = \ln (a / b)\) to get \(\ln ((x^4y^7) / z^3)\). Thus, the expression is now a single logarithm.

Key Concepts

Logarithmic Expression

A logarithmic expression represents an equation that involves logarithms. Logarithms are useful tools in mathematics that help us solve equations that involve exponential relationships, as they allow us to 'unpack' exponents. In essence, a logarithm answers the question: "To what power must a given base be raised, to produce a specific number?" If you see something like \( \log_b(x)\), it means you are searching for the power you need to raise the base \(b\) to achieve \(x\).

• When you condense a logarithmic expression, you use properties of logarithms to combine multiple terms into a single, simpler formula.
• The objective is often to express everything as one log term with a coefficient of 1 or in a form that's easy to work with.

This condensing technique is practical in solving complex logarithmic problems, simplifying calculations, or working towards a solution where exact calculations may not be desired.

Logarithm of a Power

The property of logarithm of a power simplifies expressions where logarithms have coefficients. It allows us to convert these coefficients into exponents on their respective terms.For example, if you encounter terms like \(4 \ln x\), you can apply this property: \[ a \ln b = \ln (b^a) \].

• This turns \(4 \ln x\) into \(\ln (x^4)\).
• Similarly, \(7 \ln y\) becomes \(\ln (y^7)\), and \(-3 \ln z\) becomes \(\ln (z^3)\).

By dealing with powers instead of coefficients, the process of combining these logarithms into a single form becomes much more straightforward. This property is a powerful tool that helps in the manipulation of logarithms in expressions.

Logarithm of a Product

After converting any coefficients into exponents using the logarithm of a power, the next property to consider is the logarithm of a product. This simplifies the sum of multiple logarithmic terms into a single logarithm of their product.According to the rule: \(\ln a + \ln b = \ln (a \cdot b)\), you can combine addition within logarithmic expressions into a multiplication inside one log term.

• So, when you have terms like \(\ln (x^4) + \ln (y^7)\), you can combine them into \(\ln (x^4y^7)\).

This property highlights how multiplication corresponds with addition within logarithmic terms, offering a simplified, elegant expression. This transformation is incredibly useful when working with large data sets or equations, easing computational efforts.

Logarithm of a Division

Finally, the property of logarithm of a division further simplifies expressions by merging subtraction into division within a logarithmic term. This is a vital step when condensing a logarithmic expression completely.The property \(\ln a - \ln b = \ln (a / b)\) helps transform the subtraction of logs into the division of their inner terms.

• If you have an expression like \(\ln (x^4y^7) - \ln (z^3)\), you apply this property to get \(\ln ((x^4y^7) / z^3)\).

By converting subtraction into division inside a log expression, you achieve the final, fully condensed form. This step is particularly efficient for simplifying complex logarithmic scenarios, making it easier to assess or calculate further. Understanding this property allows you to manage logarithmic functions in both academic and real-world calculus scenarios effectively.