Problem 260 - Precalculus

Question

For the following exercises, condense to a single logarithm if possible. $$ \ln (a)-\ln (d)-\ln (c) $$

Step-by-Step Solution

Answer

\( \ln\left(\frac{a}{d \cdot c}\right) \)

Step 1: Understand Logarithm Properties

Recall the logarithmic property that states the difference of two logarithms can be expressed as the logarithm of a quotient: \( \ln(x) - \ln(y) = \ln\left(\frac{x}{y}\right) \). This will help us simplify the given expression into a single logarithm.

Step 2: Apply the Properties to First Two Terms

Start by applying the property to the first two terms: \( \ln(a) - \ln(d) \). Using the quotient property results in \( \ln\left(\frac{a}{d}\right) \).

Step 3: Integrate the Third Logarithm

Now integrate the third term \( -\ln(c) \). Again, use the quotient property: \( \ln\left(\frac{a}{d}\right) - \ln(c) = \ln\left(\frac{a/d}{c}\right) \).

Step 4: Final Condensed Expression

Combine the terms into a single logarithm: \( \ln\left(\frac{a}{d \cdot c}\right) \). This is the final condensed form of the original expression.

Key Concepts

Condensing Logarithms

Condensing logarithms essentially means transforming an expression made up of multiple logarithms into a single logarithmic expression. This process is immensely helpful in simplifying complex logarithmic equations and making cumbersome expressions easier to work with.

To execute this, you’ll need to have a strong grasp of logarithmic properties and operations, such as the power, product, and quotient rules. By employing these rules, we can bring together separate logarithms into one cohesive unit.

In our exercise, we aim to condense the expression \( \ln(a) - \ln(d) - \ln(c) \). We do this by using the properties of logarithms, specifically focusing on the quotient property to reduce the complexity by combining terms.

Quotient Property

The quotient property of logarithms is a powerful tool that allows us to take the difference of two logarithms and rewrite it as a logarithm of a quotient.

Let's dive into what this means: when you have an expression like \( \ln(x) - \ln(y) \), this can be rewritten using the quotient property as \( \ln\left(\frac{x}{y}\right) \).

• This is extremely valuable because it allows us to simplify expressions significantly.
• In addition, it helps in reducing the number of logarithmic terms.

Applying this property to our case, we started by combining \( \ln(a) \) and \( \ln(d) \). Using the quotient property, we formulated it into \( \ln\left(\frac{a}{d}\right) \).

Once we had this, we applied the same logic again for the next term, integrating \( \ln(c) \), to finally get the single logarithm \( \ln\left(\frac{a}{d \cdot c}\right) \). This showcases the simplicity and utility offered by the quotient property.

Logarithmic Expressions

Logarithmic expressions involve the use of logarithms to represent relationships or calculations, often involving exponential terms.

Here's a clear understanding of what classifies as a logarithmic expression:

• It includes terms that are in the form of logarithms (e.g., \( \ln(x) \)).
• Such expressions can be simplified, expanded, or condensed using logarithmic rules.
• Logarithmic expressions can appear in various forms, for example, involving sums, differences, or products of logs.

When working with these expressions, you often need to condense or expand them depending on the requirement.

The exercise demonstrated how to manage a difference of logarithms by converting it into a single term. This not only simplifies the expression but also enhances your ability to analyze and manipulate logarithmic equations efficiently. Understanding these transformations can be crucial for solving various mathematical problems and analyses.