Problem 47 - Precalculus

Question

Write each expression as a single logarithm. $$\frac{1}{2} \ln (x+3)-\frac{1}{3} \ln (x+2)-\ln (x)$$

Step-by-Step Solution

Answer

\( \ln\left(\frac{\sqrt{x+3}}{x\sqrt[3]{x+2}}\right) \)

Step 1: Apply Logarithm Power Rule

The power rule for logarithms states that \( a \ln(b) = \ln(b^a) \). We will apply this to each term in the expression:1. \( \frac{1}{2} \ln(x+3) = \ln((x+3)^{1/2}) = \ln(\sqrt{x+3}) \)2. \( \frac{1}{3} \ln(x+2) = \ln((x+2)^{1/3}) = \ln(\sqrt[3]{x+2}) \)

Step 2: Combine Using Logarithm Quotient Rule

The expression is now \( \ln(\sqrt{x+3}) - \ln(\sqrt[3]{x+2}) - \ln(x) \). The quotient rule for logarithms states \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). We apply this to combine the terms:1. \( \ln(\sqrt{x+3}) - \ln(\sqrt[3]{x+2}) = \ln\left(\frac{\sqrt{x+3}}{\sqrt[3]{x+2}}\right) \)2. \( \ln\left(\frac{\sqrt{x+3}}{\sqrt[3]{x+2}}\right) - \ln(x) = \ln\left(\frac{\sqrt{x+3}}{x\sqrt[3]{x+2}}\right) \)

Step 3: Simplify and Finalize Expression

The expression \( \ln\left(\frac{\sqrt{x+3}}{x\sqrt[3]{x+2}}\right) \) is the final, simplified single logarithm. It represents the original expression combined into one logarithmic term.

Key Concepts

Logarithm Power Rule

To begin with, let's dive into the logarithm power rule, which is an essential tool when dealing with logarithmic expressions that have coefficients. The rule can be stated as: if you have an expression like \( a \ln(b) \), you can rewrite it as \( \ln(b^a) \). This rule is helpful because it allows you to transform products or coefficients of logarithms into more manageable forms that involve powers.

Consider the expression \( \frac{1}{2} \ln(x+3) \). Using the power rule, you can convert this to a simpler form: \( \ln((x+3)^{1/2}) \) or just \( \ln(\sqrt{x+3}) \). Similarly, \( \frac{1}{3} \ln(x+2) \) becomes \( \ln((x+2)^{1/3}) \) or \( \ln(\sqrt[3]{x+2}) \).

By doing this, we eliminate the fractions in front of the logarithms and turn them into expressions involving roots. This makes the logarithms easier to combine in later steps, using other logarithmic rules.

Logarithm Quotient Rule

After applying the power rule, the expressions can look a bit more complex, but don't worry—this is where the logarithm quotient rule comes in handy. The quotient rule is expressed as: \( \ln(a) - \ln(b) = \ln\left(\frac{a}{b}\right) \). This rule allows us to turn a subtraction of logarithms into a division within a single logarithm.

Let's apply this rule to the expression \( \ln(\sqrt{x+3}) - \ln(\sqrt[3]{x+2}) - \ln(x) \). The first two terms can be combined using the quotient rule to become \( \ln\left(\frac{\sqrt{x+3}}{\sqrt[3]{x+2}}\right) \). Now, apply the quotient rule again with the next term, \( \ln(x) \), resulting in \( \ln\left(\frac{\sqrt{x+3}}{x\sqrt[3]{x+2}}\right) \).

This transformation is powerful because it allows us to simplify complex logarithmic expressions into a single logarithm, making the math much easier to handle.

Combining Logarithms

Finally, we arrive at the key concept of combining logarithms. Once you have applied both the power rule and the quotient rule, your expression has significantly reduced in complexity. The entire process relies on rewriting and simplifying multiple logarithmic parts into a single expression.

In our example, the final expression \( \ln\left(\frac{\sqrt{x+3}}{x\sqrt[3]{x+2}}\right) \) is the culmination of combining all initial parts into one concise logarithmic term. Thereby, multiple logarithms have been seamlessly merged according to the rules we've discussed earlier. This approach not only looks tidier but is also more functional when solving further problems.

By mastering these processes, you can tackle various complex logarithmic expressions with confidence, providing you with a strong foundation for handling logarithms in any mathematical context.