Problem 51 - Essential Calculus Early Transcendentals

Question

Express the given quantity as a single logarithm. $$ \ln 5+5 \ln 3 $$

Step-by-Step Solution

Answer

The expression simplifies to \( \ln 1215 \).

Step 1: Understand the Properties of Logarithms

Before simplifying the expression, recall some key properties of logarithms. Particularly, 1. The **product property**: \( \ln(a) + \ln(b) = \ln(ab) \).2. The **power property**: \( n \ln(x) = \ln(x^n) \).These properties will help combine and simplify logarithmic terms.

Step 2: Apply the Power Property

Looking at the expression \( \ln 5 + 5 \ln 3 \), notice the term \( 5 \ln 3 \). Use the power property of logarithms to rewrite this term as a single logarithm: \[ 5 \ln 3 = \ln(3^5) = \ln(243) \].Thus, the expression now becomes: \( \ln 5 + \ln 243 \).

Step 3: Apply the Product Property

Now that the expression is \( \ln 5 + \ln 243 \), apply the product property of logarithms to combine these into a single logarithm: \[ \ln 5 + \ln 243 = \ln(5 \times 243) = \ln(1215) \]. Here, 5 multiplied by 243 results in 1215.

Key Concepts

Understanding the Product Property

The product property of logarithms is a powerful tool that allows us to combine two logarithmic expressions, provided they have the same base. This property states that the logarithm of a product is equal to the sum of the logarithms. In mathematical terms, this is expressed as

• \( \ln(a) + \ln(b) = \ln(ab) \)

When you're given an expression involving the sum of two logarithms, you can apply this property to rewrite the expression as a single logarithm.

For example, if you have the expression \( \ln 5 + \ln 243 \), using the product property will result in \( \ln(5 \times 243) \). This allows for a simplified form of the logarithmic expression, making calculations smoother and easier to handle.

Exploring the Power Property

The power property is another crucial concept in logarithms. It enables us to bring a coefficient outside the logarithm as an exponent within the logarithmic function. This property is written mathematically as

• \( n \ln(x) = \ln(x^n) \)

This means that if you have a numeric coefficient in front of a logarithm, it can be rewritten as an exponent. This can make calculations and simplifications much less cumbersome.

For instance, with the expression \( 5 \ln 3 \), we apply the power property to get \( \ln(3^5) \). By converting the expression this way, we can simplify further calculations, like combining it with other logarithmic terms using the product property.

Simplifying Expressions Using Logarithm Properties

Simplifying expressions involving logarithms often requires applying both the product and power properties. The goal is to consolidate multiple terms into a single, more manageable logarithmic expression. Let's break down the simplification process using these properties.

First, identify any multipliers outside a logarithm and apply the power property.

• Suppose you have \( 5 \ln 3 \). Rewrite this using the power property as \( \ln(3^5) \).

Next, look for logarithmic terms that can be combined using the product property.

• For \( \ln 5 + \ln 243 \), combine them into one expression: \( \ln(5 \times 243) = \ln(1215) \).

Applying these steps allows for the expression to be expressed more simply and usually in a form that is more straightforward to work with in further calculations or problem-solving scenarios.