Problem 50 - 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. $$ \frac{1}{2} \ln x+\ln y $$

Step-by-Step Solution

Answer

The condensed expression is \(\ln (x^{1/2}y)\)

Step 1: Apply the Power Rule

The first part of the expression is \(\frac{1}{2} \ln x\), which can be rewritten using the Power Rule as \(\ln (x^{1/2})\). This simplifies the given expression to: \(\ln (x^{1/2}) + \ln y\)

Step 2: Apply the Product Rule

Now that the expression is a sum of logs, it can be rewritten as a single logarithm using the Product Rule. This rule states that the sum of two logs is the log of their product. Apply this rule to get the final solution: \(\ln (x^{1/2}y)\)