Problem 37 - 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. $$ \ln \left[\frac{x^{3} \sqrt{x^{2}+1}}{(x+1)^{4}}\right] $$

Step-by-Step Solution

Answer

The expanded logarithmic expression is: \( 3\ln(x) + \frac{1}{2}\ln(x^2+1) - 4\ln(x+1) \)

Step 1: Apply the Quotient Property Logarithm

The quotient property of logarithm states that the logarithm of a quotient equals the difference of the logarithms. So start by applying the quotient property to the function: \( \ln \left[\frac{x^{3} \sqrt{x^{2}+1}}{(x+1)^{4}}\right] = \ln (x^3 \sqrt{x^2+1}) - \ln((x+1)^4) \)

Step 2: Apply the Product Property Logarithm

Next, apply the product property, which states that the logarithm of a product is equal to the sum of the logarithms of the numbers being multiplied together: \( \ln (x^3 \sqrt{x^2+1}) - \ln((x+1)^4) = \ln(x^3) + \ln(\sqrt{x^2+1}) - \ln((x+1)^4) \)

Step 3: Apply the Power Property Logarithm

After that, apply the power property for all terms. The power property states that for any positive number x and integer n, the equation \(\ln x^n = n \ln x\) holds. Therefore, apply this property to all terms: \( \ln(x^3) + \ln(\sqrt{x^2+1}) - \ln((x+1)^4) = 3\ln(x) + \frac{1}{2}\ln(x^2+1) - 4\ln(x+1) \)

Step 4: Final expression

Upon using all of these properties of logarithms, the final, simplified expression is: \( 3\ln(x) + \frac{1}{2}\ln(x^2+1) - 4\ln(x+1) \)