Problem 61 - 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. $$ 3 \ln x+5 \ln y-6 \ln z $$

Step-by-Step Solution

Answer

The expression condensed to a single logarithm with a coefficient of 1 is \(\ln(x^3y^5 / z^6)\)

Step 1: Apply power rule

Applying the logarithmic power rule, the given expression becomes: \(\ln(x^3) + \ln(y^5) - \ln(z^6)\)

Step 2: Apply addition and subtraction rule

Next, the addition and subtraction rules for logarithms are applied. This leads to:\(\ln(x^3y^5 / z^6)\)

Step 3: Maintain coefficient of 1

Since we need a single logarithm with a coefficient of 1, the expression maintains its current state. It is already simplified as per the problem's requirement.

Key Concepts

Properties of Logarithms

Logarithms have certain properties that make it easier to manipulate and simplify expressions. Understanding these properties is key when working with complex logarithmic equations.
These properties include the power rule, product rule, and quotient rule, which help to combine, expand, or simplify logarithmic terms.

• Power Rule: States that \( \log_b(a^c) = c \log_b(a) \)
• Addition Rule (Product Rule): States that \( \log_b(m) + \log_b(n) = \log_b(m \cdot n) \)
• Subtraction Rule (Quotient Rule): States that \( \log_b(m) - \log_b(n) = \log_b\left( \frac{m}{n} \right) \)

These rules help condense multiple logarithmic terms into a single, more manageable expression.

Power Rule

The power rule is a fundamental property of logarithms that helps simplify expressions by moving an exponent in front of the log.
For example, in the expression \(3 \ln x\), the exponent 3 can be "brought down" using the power rule, resulting in \( \ln(x^3) \).

• Initially, you have \( c \log_b(a) \).
• The power rule allows you to write it as \( \log_b(a^c) \).

Using the power rule can greatly simplify the computation of logarithmic expressions, making them more straightforward to evaluate.

Addition Rule

The addition rule, also known as the product rule, states that the sum of two logarithms is equivalent to the logarithm of the product of their arguments.
This rule is particularly useful when dealing with expressions where logarithms are added together.

• For example, \( \ln(x^3) + \ln(y^5) \) can be rewritten using the addition rule as \( \ln(x^3 \cdot y^5) \).
• This helps condense multiple terms into one, reducing the complexity of the expression.

Understanding the addition rule is essential for combining terms and simplifying log equations.

Subtraction Rule

The subtraction rule, or the quotient rule, is another core property of logarithms. It helps to simplify expressions where logarithms are being subtracted.
According to this rule, subtracting one logarithm from another is equivalent to taking the logarithm of the quotient of their arguments.

• The expression \( \ln(x^3y^5) - \ln(z^6) \) becomes \( \ln\left(\frac{x^3y^5}{z^6}\right) \).
• This effectively condenses the fractional expression within the logarithm, simplifying the overall equation.

Mastering the subtraction rule allows you to streamline logarithmic calculations and arrive at solutions more efficiently.