Problem 55 - 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. \(5 \ln x-2 \ln y\)

Step-by-Step Solution

Answer

The simplified form of the given expression \(5 \ln x - 2 \ln y\) is \(\ln (x^5 / y^2)\)

Step 1: Applying the Power Rule

Apply the power rule of logarithms which states, \(a \ln b = \ln (b^a)\). This can be applied to both \(5 \ln x\) and \(-2 \ln y\) separately to get the new expression: \(\ln (x^5) - \ln (y^2)\)

Step 2: Applying the Quotient Rule

Next, apply the quotient rule of logarithms which states, \(\ln a - \ln b = \ln( a / b)\). Apply this rule to the modified expression to get: \(\ln (x^5 / y^2)\)

Step 3: Final Simplification

Now the expression has been converted to a single logarithm with no coefficient, which is the final answer.

Key Concepts

Properties of Logarithms

Logarithms transform complex multiplication and division problems into simpler addition and subtraction tasks. Several properties make this possible.

• Product Rule: This rule tells us that \( \ln(a) + \ln(b) = \ln(ab) \). In other words, when multiplying numbers inside a logarithm, you can add the logs instead.
• Quotient Rule: This states that \( \ln(a) - \ln(b) = \ln(a/b) \). For division, you subtract one log from the other.
• Power Rule: It's represented by \( a \ln(b) = \ln(b^a) \). You can turn the coefficient of a log into an exponent.

These rules are essential for simplifying logarithmic expressions and solving logarithmic equations. They help you condense or expand logs into simpler forms.

Power Rule

The Power Rule is a handy property for simplifying logarithms that come with coefficients.
To use this rule, you should transform the coefficient of the log expression into an exponent. For example, in the term \( 5 \ln x \), the coefficient 5 becomes an exponent to give \( \ln(x^5) \).
Similarly, when you have \(-2 \ln y\), it changes to \( \ln(y^{-2}) \). This conversion makes both logs easier to manage and allows further simplifications. It prepares logarithmic expressions for the Quotient Rule, which comes next.

Quotient Rule

The Quotient Rule simplifies expressions involving the difference of two logarithms.
By using this rule, you can turn \( \ln(a) - \ln(b) \) into \( \ln(a/b) \). It essentially means that you can divide numbers inside the log function instead of dealing with subtraction outside.
In the exercise given, \( \ln(x^5) - \ln(y^2) \) gets condensed into \( \ln(x^5 / y^2) \). The expression is now a single logarithm, indicating successful simplification using the quotient property.

Single Logarithm

Turning multiple logarithm expressions into a single logarithm is often the goal to simplify problems. Through properties like the Power and Quotient Rules, log expressions can be condensed.
The primary aim is to have no extra coefficients and result in an expression such as \( \ln(x/y) \) rather than \( a \ln(x) - b \ln(y) \). In the exercise, by applying these rules, \( 5 \ln(x) - 2 \ln(y) \) becomes \( \ln(x^5 / y^2) \). This form is easy to interpret and can be calculated in simpler steps.
A single logarithm is often easier to evaluate or use in further calculations. This is why learning these properties is crucial to mastering logarithmic problems.