Problem 59 - 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. \(4 \ln (x+6)-3 \ln x\)

Step-by-Step Solution

Answer

\(\ln \frac{(x+6)^4}{x^3}\)

Step 1: Apply the Power Rule

The power rule is applied to the logarithmic terms to remove the coefficients. Application of the power rule turns the expression into \(\ln (x+6)^4 - \ln x^3\).

Step 2: Apply the Quotient Rule

The quotient rule is applied next to the expression derived from step 1. According to the quotient rule, the subtraction of logarithms can be replaced by the division of the initial numbers. This results in the expression: \(\ln \frac{(x+6)^4}{x^3}\).

Step 3: Final Answer and Evaluation

The logarithm is now written as a single logarithm with a coefficient one, which satisfies the conditions of the exercise. Check whether the resulting expression can be further simplified without a calculator. In this case, no simplification is possible, hence the final result is \(\ln \frac{(x+6)^4}{x^3}\).

Key Concepts

Power Rule in Logarithms

The power rule is a nifty little trick in the world of logarithms. It helps us manage and simplify expressions beautifully. Generally, it states that you can bring a coefficient in front of a logarithm as an exponent inside the logarithm. This means, if you have a coefficient in front of a logarithm, you can move it up as an exponent of the term inside the logarithm.
For any given logarithmic expression say, \(a \ln(b)\), where \(a\) and \(b\) are any numbers, you can rewrite it using the power rule as \(\ln(b^a)\).
This transformation is immensely helpful when dealing with multiple logarithmic terms in an equation. By converting them into a simpler form, it becomes easier to combine these terms later on.
For instance, in our original exercise, applying the power rule helps change \(4 \ln(x+6)\) into \(\ln(x+6)^4\). Similarly, \(3 \ln x\) becomes \(\ln x^3\). This sets the stage perfectly for further operations like applying the quotient rule.

Quotient Rule in Logarithms

After transforming the expressions using the power rule, we can move on to the quotient rule of logarithms. The quotient rule helps us deal with the subtraction of logarithms.
When you subtract one logarithm from another, this rule beautifully transforms the operation into a division operation within a single logarithmic term.
The rule states that \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right)\).
Applying this rule smartly simplifies our tasks. Instead of working with multiple expressions, we reduce them to just one. In the context of our example, we apply the quotient rule to get \(\ln\left(\frac{(x+6)^4}{x^3}\right)\).
This means we have simplified a potentially complex issue into something that’s a lot more manageable. This one-logarithm view is especially helpful when solving equations or working under specific conditions, like writing the expression as one logarithm with a coefficient of 1.

Properties of Logarithms

Understanding the properties of logarithms is crucial for simplifying expressions and solving logarithm-based problems. These properties like the product rule, quotient rule, and power rule make manipulating logarithms easier.
The power rule and quotient rule, which we have already discussed, are among these properties. Here's a quick recap:

• Power Rule: \(\ln(a^b) = b \cdot \ln(a)\).
• Quotient Rule: \( \ln(\frac{a}{b}) = \ln(a) - \ln(b)\).

The properties of logarithms allow you to condense multiple logarithmic expressions into a single expression or vice versa. They are essential whenever simplification is possible or needed.
These properties are not just formulas to memorize; they are tools that, when applied effectively, can change a seemingly complex problem into one that’s straightforward. For students, mastering these rules is key to tackling not just exercises, but a variety of computational problems across algebra and calculus.