Problem 43 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to combine the expression. $$ 4 \log x-\frac{1}{3} \log \left(x^{2}+1\right)+2 \log (x-1) $$

Step-by-Step Solution

Answer

\(\log\left(\frac{x^4 (x - 1)^2}{(x^2 + 1)^{1/3}}\right)\)

Step 1: Apply the Power Rule

The power rule of logarithms states that \( a \cdot \log b = \log(b^a) \). Apply this rule to each term in the expression: \(4 \log x\) becomes \(\log(x^4)\); \(\frac{1}{3} \log(x^2 + 1)\) becomes \(\log((x^2 + 1)^{1/3})\); and \(2 \log(x - 1)\) becomes \(\log((x - 1)^2)\).

Step 2: Combine Using the Product Rule

The product rule of logarithms states that \( \log a + \log b = \log(ab) \). Apply this rule to combine the terms \(\log(x^4)\) and \(\log((x - 1)^2)\): \(\log(x^4) + \log((x - 1)^2) = \log(x^4 (x - 1)^2)\).

Step 3: Combine Using the Quotient Rule

The quotient rule states that \( \log a - \log b = \log(\frac{a}{b}) \). Subtract \(\log((x^2 + 1)^{1/3})\) from the combined expression from Step 2: \(\log(x^4 (x - 1)^2) - \log((x^2 + 1)^{1/3}) = \log\left(\frac{x^4 (x - 1)^2}{(x^2 + 1)^{1/3}}\right)\).

Step 4: Write the Final Combined Expression

Now write the final combined expression: \(\log\left(\frac{x^4 (x - 1)^2}{(x^2 + 1)^{1/3}}\right)\).

Key Concepts

Power Rule

The Power Rule of logarithms is a useful tool for simplifying expressions containing logarithms with coefficients. The rule tells us that if you have a multiplier in front of a logarithmic expression, you can bring this number up as a power of the argument inside the log. The formula for this rule is given by:

• \( a \cdot \log b = \log(b^a) \)

This transformation makes it easier to manipulate and combine logarithmic expressions. For instance, if you have \(4 \log x\), you can apply the power rule to convert it to \(\log(x^4)\).
Similarly, \(\frac{1}{3} \log(x^2 + 1)\) becomes \(\log((x^2 + 1)^{1/3})\). This step of rewriting is crucial for further simplification using other logarithmic rules.
Keep in mind that simplifying inside the logarithm by expressing it as a power can help when combining multiple logarithms.

Product Rule

The Product Rule is another fundamental law of logarithms which helps us combine logarithmic expressions with addition. It states that the logarithm of a product is equal to the sum of the logarithms of its factors. Written formally:

• \( \log a + \log b = \log(ab) \)

Understanding this rule allows you to merge separate logarithmic terms into a single term. For example, if you have \(\log(x^4) + \log((x - 1)^2)\), you can combine them using the product rule to form \(\log(x^4 (x - 1)^2)\).
This process makes it possible to simplify more complex expressions, particularly when multiple terms are involved.
The product rule is optimal for situations where you have an addition between logarithms and you want to express them as a single logarithm of a product.

Quotient Rule

The Quotient Rule of logarithms is a powerful technique for combining logarithms through subtraction. It states that when you subtract one logarithm from another, you are essentially taking the logarithm of the quotient of the two original arguments:

• \( \log a - \log b = \log\left(\frac{a}{b}\right) \)

This rule is particularly useful when dealing with expressions that have subtracted logarithms needing to be simplified.
In the given exercise, after combining terms using the product rule, the expression becomes \(\log(x^4(x - 1)^2)\).
Then, by utilizing the quotient rule, we can subtract \(\log((x^2 + 1)^{1/3})\) from this combined expression, resulting in \(\log\left(\frac{x^4 (x - 1)^2}{(x^2 + 1)^{1/3}}\right)\).
This final step reflects how subtraction in the logarithmic domain translates to division of terms within the argument of the log function.