Problem 55 - Algebra and Trigonometry

Question

Use properties of logarithms to condense each 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

After applying the coefficient rule and then the quotient rule for logarithms, the expression simplifies to \(\ln \left(\frac{x^5}{y^2}\right)\)

Step 1: Apply coefficient rule

Apply the coefficient rule of logarithms \(a \ln b = \ln (b^a) \) to the given expression. This rule says that coefficient of the logarithm can be moved as a power to the number it acts upon. The equation becomes:\[\ln (x^5) - \ln (y^2)\]

Step 2: Apply quotient rule

Apply the quotient rule of logarithms \(\ln a - \ln b = \ln (a/b)\). This rule allows us to combine two logarithmic terms, subtracting the second value from first in a single logarithm. Applying this rule gives:\[\ln \left(\frac{x^5}{y^2}\right)\]

Key Concepts

Coefficient Rule

The coefficient rule is a fundamental property of logarithms that simplifies expressions by moving the coefficient of a logarithm to be an exponent of its argument. In other words, if you have a logarithm with a coefficient, such as \(a \ln b\), this is equivalent to \(\ln(b^a)\).
This transformation makes it easier for you to work with the expression since you are reducing the number of terms you have to deal with into a single logarithmic function.

• Suppose you have \(5 \ln x\). By applying the coefficient rule, \(5\) moves to the exponent, so it becomes \(\ln(x^5)\).
• Similarly, \(2 \ln y\) translates to \(\ln(y^2)\) when you use the coefficient rule.

Practicing this rule helps in preparing you to combine logarithmic terms efficiently and supports better understanding when more complex expressions are encountered.

Quotient Rule

The quotient rule for logarithms helps you combine two logarithmic expressions into one by using division. This rule states that the difference between two logarithms, say \( \ln a - \ln b\), can be represented as a single logarithm with a quotient inside: \(\ln(\frac{a}{b})\).
By applying this rule, you can simplify complex expressions involving differences of logarithms.

• For instance, use this rule to combine \(\ln(x^5)\) and \(\ln(y^2)\). Since there is a subtraction, you write this as \(\ln(\frac{x^5}{y^2})\).

This helps in "condensing" the logarithmic expression, making it more compact and often easier to evaluate or further manipulate.

Condensing Logarithms

Condensing logarithms involves using the properties of logarithms to combine multiple terms into a single logarithm. By employing the coefficient and quotient rules together, you can transform expressions to have fewer, more manageable terms.
This technique is crucial because it simplifies the work needed to perform further algebraic operations and can clarify the results.
In our example, starting with \(5 \ln x - 2 \ln y\), apply the coefficient rule to get \(\ln(x^5) - \ln(y^2)\).
Then, use the quotient rule to condense the expression to \(\ln(\frac{x^5}{y^2})\).

• These steps ensure the expression is simplified into a single logarithm.
• This simplification can often allow you to see what needs to be done next, whether that's solving the equation or evaluating further.

Understanding how to condense logarithms can make complex problems seem more straightforward and less intimidating.