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

\(\ln \left[\frac{x^{4} \sqrt{x^{2}+3}}{(x+3)^{5}}\right] = 4\ln(x) + \frac{1}{2}\ln(x^{2}+3) - 5\ln(x+3) \)

Step 1: Apply the quotient rule

Apply the quotient rule of logarithms, which states that the logarithm of a quotient is the difference of the logarithms. In this case, we get: \(\ln(x^{4}\sqrt{x^{2}+3}) - \ln((x+3)^{5})\)

Step 2: Apply the product rule

Apply the product rule to the first term involving the natural logarithm, which states that the logarithm of a product is the sum of the logarithms. Applying this rule here gives: \(\ln(x^{4}) + \ln(\sqrt{x^{2}+3}) - \ln((x+3)^{5})\)

Step 3: Apply the power rule

Next, apply the power rule which states that the logarithm of a number raised to an exponent is equal to that exponent times the logarithm of the number. Applying this rule here gives: \(4\ln(x) + \frac{1}{2}\ln(x^{2}+3) - 5\ln(x+3)\)

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is an essential property in simplifying and solving logarithmic expressions. It states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. This can be expressed as:

• \( \log_b \left( \frac{M}{N} \right) = \log_b(M) - \log_b(N) \)

Let's see how this rule was applied in the solution. The given expression is \( \ln \left[ \frac{x^4 \sqrt{x^2+3}}{(x+3)^5} \right] \). By using the quotient rule, we simplify it to:

• \( \ln(x^4 \sqrt{x^2+3}) - \ln((x+3)^5) \)

We separate the logarithm of the numerator \( x^4 \sqrt{x^2+3} \) from the logarithm of the denominator \( (x+3)^5 \). This step is crucial as it allows further simplification using other logarithmic rules.

Product Rule of Logarithms

Once we have dealt with the quotient, we often encounter products within the expressions. The product rule of logarithms helps here. It states that the logarithm of a product is the sum of the logarithms of the factors. This can be mathematically expressed as:

• \( \log_b(M \cdot N) = \log_b(M) + \log_b(N) \)

After applying the quotient rule in the previous step, the expression \( \ln(x^4 \sqrt{x^2+3}) \) can be further simplified using the product rule:

• \( \ln(x^4) + \ln(\sqrt{x^2+3}) \)

We segregate the product within the logarithm into its components. This separation is instrumental for using other logarithmic properties to make the expression simpler or evaluable.

Power Rule of Logarithms

The power rule of logarithms further simplifies expressions involving exponents. This rule states that the logarithm of a power is the exponent times the logarithm of the base. It can be represented as:

• \( \log_b(M^p) = p \cdot \log_b(M) \)

In the context of our problem, we applied the power rule to the terms already expanded by the quotient and product rules:

• \( \ln(x^4) \) becomes \( 4 \ln(x) \)
• \( \ln((x+3)^5) \) becomes \( 5 \ln(x+3) \)
• \( \ln(\sqrt{x^2+3}) \) becomes \( \frac{1}{2} \ln(x^2+3) \)

Here, each exponent is brought down as a multiplier in front of its respective logarithm, reducing further complexity in the expression. Using these methods in tandem simplifies complex expressions and highlights the efficiency of logarithmic rules. By breaking down each part, the original expression becomes manageable and easier to interpret.