Problem 34 - Algebra and Trigonometry

Question

\(19-44\) Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{a^{2}}{b^{4} \sqrt{c}}\right) $$

Step-by-Step Solution

Answer

The expanded form is \(2 \cdot \log(a) - 4 \cdot \log(b) - \frac{1}{2} \cdot \log(c)\).

Step 1: Apply the Quotient Rule

The Logarithm Quotient Rule states that \( \log\left(\frac{M}{N}\right) = \log(M) - \log(N) \). Using this rule, the expression can be rewritten as:\[\log \left(\frac{a^{2}}{b^{4} \sqrt{c}}\right) = \log(a^{2}) - \log(b^{4} \sqrt{c})\]

Step 2: Break Down the Denominator Using the Product Rule

The Logarithm Product Rule states that \( \log(MN) = \log(M) + \log(N) \). We can apply this to the denominator:\[\log(b^{4} \sqrt{c}) = \log(b^{4}) + \log(\sqrt{c})\]

Step 3: Simplify Using the Power Rule

The Power Rule states that \( \log(M^p) = p \cdot \log(M) \). Apply this rule to each part of the expression:1. \( \log(a^{2}) = 2 \cdot \log(a) \)2. \( \log(b^{4}) = 4 \cdot \log(b) \)3. \( \log(\sqrt{c}) = \log(c^{1/2}) = \frac{1}{2} \cdot \log(c) \)

Step 4: Combine All Components

Substitute the simplified components back into the original expansion:\[2 \cdot \log(a) - (4 \cdot \log(b) + \frac{1}{2} \cdot \log(c))\]Finally, distribute the negative sign:\[= 2 \cdot \log(a) - 4 \cdot \log(b) - \frac{1}{2} \cdot \log(c)\]

Key Concepts

Quotient Rule

The Quotient Rule for logarithms is a handy tool that simplifies the logarithm of a division. Imagine you have a logarithm with a fraction inside. Normally, dealing with fractions can be tricky, but the Quotient Rule makes it easier by breaking it into two separate logarithms. This rule states that the logarithm of a quotient is equal to the difference between the logarithms of the numerator and the denominator. For instance, if you have \(\log\left(\frac{M}{N}\right)\), it becomes \(\log(M) - \log(N)\).

When you apply the Quotient Rule, you take the entire numerator and denominator separately. This separation can make further simplification easier, as each piece can be handled on its own rather than as a part of a fraction.

Using the law step-by-step helps simplify expressions into more manageable pieces.

Product Rule

The Product Rule for logarithms is particularly helpful when dealing with products inside a logarithm. It states that the logarithm of a product of two numbers is the sum of the logarithms of those numbers. That is, \(\log(MN) = \log(M) + \log(N)\).

This rule turns multiplication inside a logarithm into addition outside, making complex expressions simpler. For example, if the expression within the logarithm is something like \(b^4 \sqrt{c}\), applying the Product Rule allows you to consider each part independently: \(\log(b^4) + \log(\sqrt{c})\).

Understanding this rule helps break down complicated expressions, especially when they consist of multiple multiplicative factors. Once simplified, each component can be handled individually and further simplified using other logarithm laws.

Power Rule

The Power Rule for logarithms allows you to bring exponents outside of the logarithm. It’s a crucial tool for simplifying logarithmic expressions where components have exponents. The Power Rule states that \(\log(M^p) = p \cdot \log(M)\). This means if you have an exponent inside the log, you can multiply that exponent with the logarithm of the base.

For example, in our expression, \(\log(a^{2})\) becomes \(2 \cdot \log(a)\), and \(\log(b^{4})\) becomes \(4 \cdot \log(b)\). Even the square root, which is a form of exponent, can be simplified: \(\log(\sqrt{c}) = \log(c^{1/2}) = \frac{1}{2} \cdot \log(c)\).

Using the Power Rule makes it much easier to manage and simplify expressions and equations involving logarithms and exponents. It effectively transforms a potentially cumbersome calculation into simpler arithmetic.