Problem 93 - College Algebra and Calculus: An Applied Approach

Question

Condense the expression to the logarithm of a single quantity.\(-\ln x-3 \ln 6\)

Step-by-Step Solution

Answer

The condensed logarithmic expression is \(\ln (216/x)\)

Step 1 - Identify and apply the power rule

The power rule states that \(a \ln b = \ln b^a\). Applying this rule to the expression \(3 \ln 6\) gives \(\ln 6^3 = \ln 216\)

Step 2 - Apply the change of sign rule

When the logarithm is negative, it can be moved to the denominator on the other side of the equation. Therefore, \(-\ln x\) becomes \(1/x\) when moved to the other side of the equation.

Step 3 - Condense the expression using the quotient rule

The quotient rule states that \(\ln a - \ln b = \ln (a/b)\). Using this rule, we combine \(\ln x\) and \(\ln 216\) to get \(\ln (216/x)\)

Key Concepts

Power Rule

The Power Rule in the context of logarithms is a convenient way to handle logarithmic expressions involving multiplication by a coefficient. This rule states that multiplying a log by a number can be converted into an exponent on the argument of the log. For example, if you have an expression in the form of \(a \ln b\), it can be rewritten as \(\ln b^a\).

This transformation helps simplify expressions, especially when you're trying to condense a logarithmic expression into a single log. In the given exercise, you encountered \(3 \ln 6\), which using the Power Rule becomes \(\ln 6^3\) or \(\ln 216\).

Essentially, moving a coefficient inside a logarithm as an exponent is an elegant way to leverage the Power Rule. It's particularly handy in solving and simplifying many forms of mathematical problems involving logarithms.

Quotient Rule

The Quotient Rule for logarithms is instrumental when you need to subtract one logarithm from another, particularly when aiming to condense expressions. This rule states that \(\ln a - \ln b = \ln \frac{a}{b}\).

In practical terms, it allows you to combine two logs with a subtraction sign into a single log with a division inside. This is crucial for simplifying complex log expressions.

In your exercise, the subtraction \(-\ln x - 3 \ln 6\) becomes \(\ln 216 - \ln x\) after applying the Power Rule. Consequently, you can use the Quotient Rule to combine these into one log: \(\ln \frac{216}{x}\).

Using the Quotient Rule efficiently can significantly streamline the process of solving log equations and make calculations much simpler.

Logarithm Properties

Understanding the core properties of logarithms can greatly assist in manipulating and simplifying logarithmic expressions. There are several useful properties, but key ones relevant to the exercise include the Power Rule, Product Rule, and Quotient Rule.

These properties provide systematic steps to simplify complex expressions. Together, they facilitate transitions between multiplication, division, and exponentiation within logarithms.

In the given problem, properties like power progression and quotienting are fundamental. They equip you with tools to either expand expressions like \( \ln b^a \) into \(a \ln b \) or condense \( \ln a - \ln b \) into a single simplified form like \(\ln \frac{a}{b}\).

Overall, a deep understanding of these properties helps immensely with both solving problems and appreciating the inherent simplicity and balance of logarithmic expressions.