Problem 49 - Precalculus Mathematics for Calculus

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 for Logarithms

The Power Rule states that \( a \log b = \log b^a \). Apply this rule to each term in the expression:- For \( 4 \log x \), use the rule to rewrite it as \( \log x^4 \).- For \( \frac{1}{3} \log (x^2+1) \), rewrite it as \( \log (x^2+1)^{1/3} \).- For \( 2 \log (x-1) \), rewrite it as \( \log (x-1)^2 \). Thus, the expression becomes:\[ \log x^4 - \log (x^2 + 1)^{1/3} + \log (x-1)^2. \]

Step 2: Use the Product Rule for Logarithms

The Product Rule for logarithms states that \( \log a + \log b = \log (ab) \). Apply this to combine the terms with addition:Combine \( \log x^4 + \log (x-1)^2 \) to get:\[ \log (x^4 (x-1)^2). \]

Step 3: Use the Quotient Rule for Logarithms

The Quotient Rule for logarithms states that \( \log a - \log b = \log \left( \frac{a}{b} \right) \). Apply this rule to combine your current expression:Subtract \( \log (x^2 + 1)^{1/3} \) from the expression obtained in Step 2:\[ \log \left( \frac{x^4 (x-1)^2}{(x^2 + 1)^{1/3}} \right). \]

Step 4: Simplify the Expression

The expression is now in its simplest combined form using the laws of logarithms:\[ \log \left( \frac{x^4 (x-1)^2}{(x^2 + 1)^{1/3}} \right). \]

Key Concepts

Laws of Logarithms

Logarithms can seem tricky at first, but understanding the core laws makes them much simpler. The three main laws of logarithms are essential in performing operations involving these functions. Here are some important points about these laws:

• The Power Rule is used when a logarithm has a coefficient in front of it, and allows us to bring this coefficient up as an exponent.
• The Product Rule helps us combine the logarithms of two multiplied numbers into a single logarithm.
• The Quotient Rule helps us write the difference of two logarithms (subtraction) as a division inside one logarithm.

Being familiar with these rules helps in simplifying complex expressions and solving equations involving logarithms. Let's go through each rule to understand them better.

Power Rule

The Power Rule is a property of logarithms that allows us to move a coefficient in front of a logarithm into the logarithm as an exponent. The rule is stated as follows:

• If you have an expression like this: \( a \log b \), you can rewrite it as \( \log b^a \).

This transformation can make calculations much more straightforward when dealing with long or complicated expressions.

For example, in the problem given, to simplify the term \( 4 \log x \), we can rewrite it using the Power Rule as \( \log x^4 \). This helps us reduce the expression to a form that's easier to combine with others using the following rules.

Product Rule

The Product Rule is used when you want to combine two logarithmic expressions that are being added. It turns addition inside logarithms into multiplication. Here's what the rule looks like:

• When you have two logarithms added together: \( \log a + \log b \), it can be combined into \( \log (ab) \).

In the exercise, applying the Product Rule helps in combining terms that were expressed in terms of addition after applying the Power Rule. For instance, the expressions \( \log x^4 \) and \( \log (x-1)^2 \) can be combined into a single term: \( \log (x^4 (x-1)^2) \). This simplifies the expression considerably before moving to the next and final important rule.

Quotient Rule

The Quotient Rule is handy when dealing with the subtraction of logarithms. It turns subtraction into division inside a single logarithm expression. Here's the rule at a glance:

• When two logarithms are subtracted: \( \log a - \log b \), you can rewrite this as a single logarithm: \( \log \left( \frac{a}{b} \right) \).

This is especially useful in reducing expressions to their simplest form. In our problem, after combining the products using the Product Rule, the result was \( \log (x^4(x-1)^2) \). Subtracting \( \log (x^2 + 1)^{1/3} \) using the Quotient Rule simplifies this into \( \log \left( \frac{x^4(x-1)^2}{(x^2 + 1)^{1/3}} \right) \). This expression is much easier to handle and represents the solution to the given problem.