Problem 56 - 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. $$ 3 \ln x-\frac{1}{3} \ln y $$

Step-by-Step Solution

Answer

The expression \(3 \ln x - \frac{1}{3} \ln y\) condenses to a single logarithm: \(\ln (x^3 / y^{1/3})\).

Step 1: Analyze the given logarithmic expression

In the expression, \(3 \ln x-\frac{1}{3} \ln y\), the logarithm is base \(e\) (since \(\ln\) is the natural logarithm) and the terms can be reduced to a single logarithm using the laws of logarithms.

Step 2: Apply the power rule to each term

Use the property \(a \log_b m = \log_b m^a\), where \(b\) is the base of the logarithm, \(a\) is the coefficient in front of the logarithm, and \(m\) is the number for which the logarithm is being taken. Apply this rule to both terms of the expression. This will make the expression become \(\ln x^3 - \ln y^{1/3}\).

Step 3: Apply the quotient rule

The quotient rule states that \(\log_b m - \log_b n = \log_b (m/n)\), where \(b\) is the base of the logarithm and \(m\) and \(n\) are the numbers for which the logarithm is being taken. The expression \(\ln x^3 - \ln y^{1/3}\) can then be written as \(\ln (x^3 / y^{1/3})\)