Problem 27 - Algebra and Trigonometry

Question

Use the Laws of Logarithms to expand the expression. $$ \log \left(\frac{x^{3} y^{4}}{z^{6}}\right) $$

Step-by-Step Solution

Answer

The expanded expression is \( 3\log(x) + 4\log(y) - 6\log(z) \).

Step 1: Apply the Quotient Rule of Logarithms

Using the quotient rule of logarithms, which states that \( \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \), we can apply this rule to the expression: \( \log \left(\frac{x^{3} y^{4}}{z^{6}}\right) = \log(x^{3} y^{4}) - \log(z^{6}) \).

Step 2: Apply the Product Rule of Logarithms

Next, use the product rule of logarithms, which states that \( \log(ab) = \log(a) + \log(b) \). Apply this rule to the expression inside the logarithm from Step 1. This gives: \( \log(x^{3} y^{4}) = \log(x^{3}) + \log(y^{4}) \).

Step 3: Apply the Power Rule of Logarithms

Now apply the power rule of logarithms, which states that \( \log(a^{b}) = b \cdot \log(a) \). Apply this rule to each term in the expression: \( \log(x^{3}) = 3 \cdot \log(x) \) and \( \log(y^{4}) = 4 \cdot \log(y) \). Also, \( \log(z^{6}) = 6 \cdot \log(z) \).

Step 4: Combine the Expanded Terms

Putting all the expanded terms together, we have: \( 3 \cdot \log(x) + 4 \cdot \log(y) - 6 \cdot \log(z) \). Thus, the expanded expression is: \( 3 \log(x) + 4 \log(y) - 6 \log(z) \).

Key Concepts

Quotient Rule of Logarithms

When dealing with logarithms, the quotient rule is a powerful tool to simplify expressions. The rule states that the logarithm of a quotient is equal to the difference of the logarithms of the numerator and the denominator. In mathematical form, it is expressed as:

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

This means if you have a fraction inside a logarithm, you can "pull apart" the fraction by subtracting the logarithm of the denominator from the logarithm of the numerator.
For example, in our original exercise, we applied the quotient rule like so: given \( \log \left( \frac{x^{3} y^{4}}{z^{6}} \right) \), we separated this into \( \log(x^{3} y^{4}) - \log(z^{6}) \). This is the first step towards simplifying complex logarithmic expressions.

Product Rule of Logarithms

The product rule of logarithms is another useful rule that helps in breaking down logarithmic expressions. According to this rule, the logarithm of a product is the sum of the logarithms of the individual factors. It can be written as:

• \( \log(ab) = \log(a) + \log(b) \)

This means that if you have a multiplication inside the logarithm, you can "split" it into the addition of separate logarithms of each multiplied value.
In our example, we applied the product rule to \( \log(x^{3} y^{4}) \), allowing us to split it into \( \log(x^{3}) + \log(y^{4}) \). This step is essential to further simplify the expression by breaking down multiplied components into simpler parts.

Power Rule of Logarithms

The power rule of logarithms is pivotal when working with exponential terms inside logarithms. This rule states that if a variable inside a logarithm is raised to a power, the power can be brought in front as a coefficient. Mathematically, it is defined as:

• \( \log(a^{b}) = b \cdot \log(a) \)

This allows us to "bring down" exponents and makes calculations simpler by converting an exponentiation into a multiplication.
In our exercise, we used this rule to transform \( \log(x^{3}) \) into \( 3 \cdot \log(x) \), \( \log(y^{4}) \) into \( 4 \cdot \log(y) \), and \( \log(z^{6}) \) into \( 6 \cdot \log(z) \). By using this rule, the expression is streamlined into simpler additive components, which are easier to handle or further manipulate if necessary.