Problem 49 - Algebra and Trigonometry

Question

\(45-54\) . 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

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

Step 1: Apply Power Rule

Apply the Power Rule of Logarithms, which states that \( n \log a = \log a^n \), to each term in the expression. For the first term, \( 4 \log x \) becomes \( \log x^4 \). For the second term, \( -\frac{1}{3} \log (x^2 + 1) \) becomes \( \log (x^2 + 1)^{-1/3} \). For the third term, \( 2 \log (x-1) \) becomes \( \log (x-1)^2 \).

Step 2: Combine Using Product Rule

Use the Product Rule of Logarithms, which states that \( \log a + \log b = \log (ab) \), to combine the logarithms. We combine \( \log x^4 \) and \( \log (x-1)^2 \) to get \( \log (x^4 (x-1)^2) \).

Step 3: Combine Using Quotient Rule

Apply the Quotient Rule of Logarithms, which states that \( \log a - \log b = \log \left(\frac{a}{b}\right) \). Combine \( \log (x^4 (x-1)^2) - \log (x^2 + 1)^{-1/3} \) to obtain \( \log \left( \frac{x^4 (x-1)^2}{(x^2 + 1)^{-1/3}} \right) \).

Step 4: Simplify the Expression

Simplify the expression from the previous step. The term with a negative exponent in the denominator is equivalent to multiplication in the numerator. Thus, \( \frac{1}{(x^2 + 1)^{-1/3}} \) becomes \( (x^2 + 1)^{1/3} \) in the numerator. The final expression is \( \log \left( x^4 (x-1)^2 (x^2 + 1)^{1/3} \right) \).

Key Concepts

Laws of Logarithms

Logarithms have rules that allow us to manipulate and simplify expressions by converting multiplication, division, and exponents into additions, subtractions, and multiplications respectively. They are incredibly useful in solving equations involving exponential relationships.

There are three primary laws of logarithms that you should be familiar with:

• The Product Rule: This allows us to take the logarithm of a product and express it as the sum of logarithms.
• The Quotient Rule: This deals with the logarithm of a division, representing it as the difference between two logarithms.
• The Power Rule: This simplifies the logarithm of a power by bringing the exponent as a multiplier in front of the logarithm.

These laws provide foundational techniques for problem-solving and are essential in calculus and algebra calculations involving exponential and logarithmic expressions.

Power Rule of Logarithms

The Power Rule of Logarithms is fundamental in mathematical manipulation. It helps convert exponential expressions into a form that is easier to manage. The rule is summarized as \( n \log a = \log a^n \).

When you see a logarithm multiplied by a number (like in our exercise with terms like \(4 \log x\)), you apply the power rule to simplify it. Instead of keeping the multiplier outside, we can move it as the exponent on the term inside the log function. Thus, \(4 \log x\) becomes \(\log x^4\). This rule is handy when merging logarithms in an expression as it allows simplifying and combining based on common bases.

Product Rule of Logarithms

The Product Rule of Logarithms allows us to write the logarithm of a product as the sum of the logarithms of its factors. This is expressed as \( \log a + \log b = \log (ab) \).

This rule is particularly useful in simplifying expressions that involve several multiplicative factors under a single logarithm. For example, if you have \( \log x^4 \) and \( \log (x-1)^2 \), you can use the Product Rule to express them as \( \log (x^4(x-1)^2) \).

This transformation facilitates the process of combining multiple logarithmic terms into one, which is often needed in further simplifications.

Quotient Rule of Logarithms

The Quotient Rule of Logarithms is used to simplify expressions where logarithms involve division. This rule says that \( \log a - \log b = \log \left( \frac{a}{b} \right) \).

In our step-by-step solution, we use the Quotient Rule to combine \( \log (x^4 (x-1)^2) - \log (x^2 + 1)^{-1/3} \). Applying the rule, these expressions become \( \log \left( \frac{x^4 (x-1)^2}{(x^2 + 1)^{-1/3}} \right) \).

This step is crucial for rewriting complex logarithmic expressions as simpler or more concise ones. It transforms subtractions of logs into divisions under a single log, simplifying calculations and making further manipulation straightforward.