Problem 41 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \sqrt{\frac{x^{2}+4}{\left(x^{2}+1\right)\left(x^{3}-7\right)^{2}}} $$

Step-by-Step Solution

Answer

\( \frac{1}{2} \log(x^{2}+4) - \frac{1}{2} \log(x^{2}+1) - \log(x^{3}-7) \)

Step 1: Apply the square root rule

The expression under the logarithm is a square root. Recall that \( \log(\sqrt{a}) = \frac{1}{2}\log(a) \). So, apply this rule to the entire expression. We have: \[ \log \sqrt{\frac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^2}} = \frac{1}{2} \log \left( \frac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^2} \right) \]

Step 2: Use the Quotient Rule

The logarithm of a quotient can be expanded using the rule \( \log(a/b) = \log(a) - \log(b) \). Apply this to the expression: \[ \frac{1}{2} \left( \log(x^{2}+4) - \log((x^{2}+1)(x^{3}-7)^2) \right) \]

Step 3: Apply the Product Rule

Next, use the logarithm product rule \( \log(ab) = \log(a) + \log(b) \) to expand the logarithm of the product in the expression: \[ \frac{1}{2} \left( \log(x^{2}+4) - (\log(x^{2}+1) + \log((x^{3}-7)^2)) \right) \]

Step 4: Apply the Power Rule

Now use the power rule \( \log(a^n) = n\log(a) \), on \( \log((x^{3}-7)^2) \):\[ \frac{1}{2} \left( \log(x^{2}+4) - (\log(x^{2}+1) + 2\log(x^{3}-7)) \right) \]

Step 5: Distribute the Factor

Finally, distribute the \( \frac{1}{2} \) throughout the expanded expression:\[ \frac{1}{2} \log(x^{2}+4) - \frac{1}{2} \log(x^{2}+1) - \log(x^{3}-7) \]

Key Concepts

Quotient Rule

The Quotient Rule is a handy tool when dealing with logarithms of divisions. If you have a fraction inside a logarithm, the Quotient Rule allows you to separate it into two simpler logarithmic expressions.
The rule tells us that

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

This rule helps to break down complex expressions into smaller and more manageable parts.
Consider the expression from the exercise:\[ \frac{1}{2} \log \left( \frac{x^{2}+4}{(x^{2}+1)(x^{3}-7)^2} \right)\]
Applying the Quotient Rule, you can expand it to:

• \( \frac{1}{2} ( \log(x^{2}+4) - \log((x^{2}+1)(x^{3}-7)^2) ) \)

By using this rule, you effectively take the complexity out of the division by expressing it as a subtraction of two logarithms. Splitting expressions into easier parts makes solving much more straightforward.

Product Rule

In situations involving products inside logarithms, the Product Rule is helpful. It's used when your expression involves multiplying two or more terms. This rule states that

• \( \log(ab) = \log(a) + \log(b) \)

Using this rule, you can turn the logarithm of a product into a sum of logarithms.
From the given exercise, the expression includes a product:\[ \frac{1}{2} ( \log(x^{2}+4) - \log((x^{2}+1)(x^{3}-7)^2) )\]
Apply the Product Rule to \( \log((x^{2}+1)(x^{3}-7)^2) \):

• \( \log(x^{2}+1) + \log((x^{3}-7)^2) \)

With this rule, multiple factors within a logarithm can be expanded into individual logarithms, simplifying the expression for further manipulation.

Power Rule

The Power Rule is specifically for dealing with exponents inside logarithms. It allows you to transform an exponent inside a logarithmic expression into a multiplier outside of it.
This rule is especially useful when part of an expression involves powers, and it is expressed as:

• \( \log(a^n) = n\log(a) \)

In the exercise, observe the term \( \log((x^{3}-7)^2) \). By applying the Power Rule, you can simplify it to:

• \( 2\log(x^{3}-7) \)

This transformation helps in managing powers effectively, enabling a more straightforward simplification process.
By breaking down even the most powerful terms to simpler logarithms, the Power Rule reduces complexity, facilitating easier calculations and understanding.