Problem 62 - 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. $$ 4 \ln x+7 \ln y-3 \ln z $$

Step-by-Step Solution

Answer

The expression can be rewritten as \( ln(x^{4}y^{7}/z^{3}) \).

Step 1: Apply the power rule of logarithms

Rewrite each logarithmic term using the power rule of logarithms: \( ln(x^{4}) + ln(y^{7}) - ln(z^{3}) \)

Step 2: Apply properties of logarithms

Next, apply the property that the logarithm of a product is the sum of the logarithms and the logarithm of a quotient is the difference of the logarithms: \( ln(x^{4}y^{7}/z^{3}) \)

Step 3: Simplify the expression

Since the problem asks you to write the expression as a single logarithm with coefficient 1, no further simplification is needed. You've already reduced the expression to a single logarithm of the form 1log(u), which is our final answer.

Key Concepts

Power Rule of Logarithms

In logarithms, the power rule is an essential tool that simplifies expressions significantly. This rule states that the logarithm of a number raised to an exponent can be expressed as the exponent multiplied by the logarithm of the base number. Mathematically, it’s written as \( \log_b (a^n) = n \cdot \log_b (a) \).

In our example, applying the power rule helps transform each term into a simpler form by bringing the coefficients inside the logarithm as exponents:

• convert \(4 \ln x\) to \(\ln(x^4)\)
• convert \(7 \ln y\) to \(\ln(y^7)\)
• convert \(3 \ln z\) to \(\ln(z^3)\)

This step is crucial in reducing the complexity of logarithmic expressions and setting up for the next rules.

Product Rule of Logarithms

The product rule of logarithms is another helpful property that simplifies the addition of logarithms. This rule states that the logarithm of a product is equal to the sum of the logarithms of individual factors. In mathematical terms, it's expressed as \( \log_b (mn) = \log_b (m) + \log_b (n) \).

In our given example, after we used the power rule to rewrite the terms in the form \(\ln(x^4) + \ln(y^7)\), we can now apply the product rule to combine them into a single logarithm: \(\ln(x^4 \cdot y^7)\). This consolidation reduces two logarithmic terms into one, thus simplifying the expression towards forming a single logarithm.

Quotient Rule of Logarithms

The quotient rule of logarithms is equally important when dealing with the difference of logarithms. This rule expresses that the logarithm of a quotient equals the difference of the logarithms: \( \log_b \left( \frac{m}{n} \right) = \log_b (m) - \log_b (n) \).

Applying this rule to our simplified expression, after the product rule, we have the expression \(\ln(x^4 \cdot y^7) - \ln(z^3)\). Using the quotient rule, we can condense this to a single logarithmic term: \(\ln\left(\frac{x^4 \cdot y^7}{z^3}\right)\).

This final step merges two terms into one, achieving the goal of expressing the logarithms as a single unit with a coefficient of 1. Each of these rules helps to streamline the expression, making them essential tools in the algebra of logarithms.