Problem 38 - College Algebra

Question

In Exercises \(1-40,\) use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \ln \left[\frac{x^{4} \sqrt{x^{2}+3}}{(x+3)^{5}}\right] $$

Step-by-Step Solution

Answer

The expanded logarithmic expression is: \( 4\ln(x) + 0.5\ln(x^{2}+3) - 5\ln(x+3) \)

Step 1: Apply the quotient rule

Let's start by applying the quotient rule of logarithms to the given expression: \( \ln \left[\frac{x^{4} \sqrt{x^{2}+3}}{(x+3)^{5}}\right] = \ln(x^{4} \sqrt{x^{2}+3}) - \ln((x+3)^{5}) \)

Step 2: Apply the power rule

Next, we apply the power rule. For \( \ln(x^{4} \sqrt{x^{2}+3}) \), it becomes \( 4\ln(x) + 0.5\ln(x^{2}+3) \). For \( \ln((x+3)^{5}) \), it becomes \( 5\ln(x+3) \): \( 4\ln(x) + 0.5\ln(x^{2}+3) - 5\ln(x+3) \)

Step 3: Apply the product rule

We do not apply the product rule because there are no multiplied terms inside any of the logarithms in our current expression.