Problem 61 - College Algebra

Question

In Exercises \(41-70,\) 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 simplified logarithmic expression is \( \ln ((x^3 y^5) / z^6) \)

Step 1: Apply the Power Rule

The power rule of logarithms states that \( b \log_a(x) = \log_a(x^b) \). By applying this rule, the given expression \( 3 \ln x + 5 \ln y - 6 \ln z \) can be rewritten as \( \ln (x^3) + \ln (y^5) - \ln (z^6) \).

Step 2: Apply the Product Rule

The product rule for logarithms states that \( \log_a(b) + \log_a(c) = \log_a(b*c) \). Apply this rule to the first two logarithms to get: \( \ln (x^3 * y^5) - \ln (z^6) \). This simplifies to \( \ln (x^3 y^5) - \ln (z^6) \)

Step 3: Apply the Quotient Rule

The quotient rule of logarithms states that \( \log_a(b) - \log_a(c) = \log_a(b/c) \). Apply this rule to condense the expression into a single logarithm: \( \ln ((x^3 y^5) / z^6) \)

Key Concepts

Power Rule

The power rule for logarithms is a handy tool when simplifying or condensing expressions involving multiple logarithms. It states that for any logarithm, \( b \log_a(x) \) can be simplified to \( \log_a(x^b) \). This means that whenever you encounter a coefficient in front of a logarithm, you can move it as an exponent of the argument inside the logarithm. This is particularly useful because it prepares the expression for further simplification using other properties.

• For example, if you have \( 3 \ln x \), it can be rewritten as \( \ln(x^3) \).
• This makes the expression easier to manage in upcoming steps.

Applying this rule in our exercise, the terms \( 3 \ln x + 5 \ln y - 6 \ln z \) are transformed into \( \ln(x^3) + \ln(y^5) - \ln(z^6) \).
Understanding the power rule simplifies and streamlines the process of condensing logarithmic expressions, as seen here.

Product Rule

After applying the power rule, the next step is to use the product rule for logarithms. This rule helps to combine logarithmic terms using multiplication. The product rule states that \( \log_a(b) + \log_a(c) = \log_a(b*c) \). This means when you have a sum of two logarithms with the same base, you can merge them into a single logarithm of the product of their arguments.

• For instance, \( \ln(x^3) + \ln(y^5) \) can be combined into \( \ln(x^3 y^5) \).
• This rule is especially useful for simplifying expressions with multiple addition terms.

In the step-by-step solution, the product rule is applied to the expression \( \ln(x^3) + \ln(y^5) - \ln(z^6) \) leading to \( \ln(x^3 y^5) - \ln(z^6) \).
This simplifies the logarithmic expression further, paving the way for using the quotient rule.

Quotient Rule

The quotient rule is the final step in our process of condensing logarithmic expressions into a single term. When you have a difference of logarithms, you can use the quotient rule to combine them. The rule is stated as \( \log_a(b) - \log_a(c) = \log_a(b/c) \). This rule tells us you can rewrite the subtraction of two logarithms as the logarithm of a division of their arguments.

• For example, \( \ln(x^3 y^5) - \ln(z^6) \) combines into a single expression \( \ln((x^3 y^5)/z^6) \).
• This is particularly effective in streamlining complex logarithmic expressions into a manageable form for further calculations or interpretations.

In our exercise, after applying the power and product rules, the quotient rule helps condense the expression into one concise log term: \( \ln((x^3 y^5)/z^6) \).
Mastery of the quotient rule, along with the power and product rules, enables efficient handling and simplification of any logarithmic expressions.