Problem 84 - Algebra 2

Question

Expand each logarithm. \(\log \frac{\sqrt{x^{2}-4}}{(x+3)^{2}}\)

Step-by-Step Solution

Answer

The expanded form of the logarithm is \(\frac{1}{2} \log (x^{2}-4) - 2 \log (x+3)\).

Step 1: Apply the Quotient Rule

First, apply the quotient rule of logarithms. This rule states that the logarithm of a quotient is the difference of the logarithms. Therefore, \(\log \frac{\sqrt{x^{2}-4}}{(x+3)^{2}}\) can be rewritten as \(\log \sqrt{x^{2}-4} - \log ((x+3)^{2})\).

Step 2: Apply the Rule of Logarithms for Exponents

Next, apply the rule of logarithms for exponents to both terms separately. So, \(\log \sqrt{x^{2}-4}\) becomes \(\frac{1}{2} \log (x^{2}-4)\), and \(\log ((x+3)^{2})\) becomes \(2 \log (x+3)\).

Step 3: Final Expression

The final expression of the logarithm becomes \(\frac{1}{2} \log (x^{2}-4) - 2 \log (x+3)\). This is the fully expanded form of the logarithm.

Key Concepts

Quotient Rule

When working with logarithmic expressions involving fractions, the quotient rule can come in very handy. The rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. This formula looks like this: \[\log\left(\frac{a}{b}\right) = \log(a) - \log(b) \]Here, we split a fraction inside a logarithm into two separate logs.

• The part \(a\) is the numerator.
• The part \(b\) is the denominator.

This rule makes it easier to manage and expand complicated logarithmic expressions.In our exercise, this rule was used to convert: \[\log \frac{\sqrt{x^{2}-4}}{(x+3)^{2}}\]into two separate parts: \[\log \sqrt{x^{2}-4} - \log((x+3)^{2}) \] This showcases how the quotient rule breaks down a log of a division into subtraction.

Exponent Rule

A key property of logarithms is the exponent rule, which allows us to manage powers inside a logarithm term more conveniently. The exponent rule states:\[\log(a^b) = b \cdot \log(a)\]Here, the exponent \(b\) comes down in front of the logarithm. It simplifies expressions by converting multiplication inside the logarithm into multiplication outside. In this exercise, this rule was applied to both terms from the expanded form.

• For \(\log \sqrt{x^2-4}\), rewriting \(\sqrt{x^2-4}\) as \((x^2-4)^{1/2}\) transformed it into:\[\frac{1}{2} \cdot \log(x^2-4)\]
• For \(\log((x+3)^{2})\), the rule changed it to \(2 \cdot \log(x+3)\).

This rule is vital for expanding logarithms since exponents are quite common in algebraic expressions.

Logarithmic Expressions

Logarithmic expressions refer to mathematical expressions that involve logarithms. These expressions are essential for simplifying and solving equations that involve exponential relationships.Working with logarithmic expressions involves knowing and applying fundamental rules like the quotient and exponent rules, which we discussed. Breaking down complex expressions into simpler components is the goal. Each component can be solved or simplified individually.

• Start with the original log expression. Use known rules to separate it into manageable parts.
• Remember, logs represent exponents. For instance, \(\log_b(a) = c\) means \(b^c = a\).
• Careful simplification is key, working through one step at a time.

Practicing these methods with various expressions will improve your confidence and skills in algebra, making complex problems seem much more conquerable.