Problem 27 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) $$

Step-by-Step Solution

Answer

The expanded form of \( \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) \) is \( 2 * \log_b{x} + \log_b{y} - 2 * \log_b{z} \).

Step 1: Apply Logarithm Quotient Rule

The first step is to apply the quotient rule. The quotient rule of logarithms states that the log of a quotient is the difference of the logs. So, \( \log _{b}\left(\frac{x^{2} y}{z^{2}}\right) \) becomes \( \log_b{x^{2} y} - \log_b(z^{2}) \).

Step 2: Apply Logarithm Product Rule

The next step involves the product rule. According to the product rule of logarithms, the log of a product is the sum of the logs. So, \( \log_b{x^{2} y} \) can be written as \( \log_b{x^{2}} + \log_b{y} \). Now the expression becomes \( \log_b{x^{2}} + \log_b{y} - \log_b(z^{2}) \).

Step 3: Apply Logarithm Power Rule

The power rule of logarithms says that the log of a quantity raised to an exponent is the product of the exponent and the log of the quantity. Therefore, \( \log_b{x^{2}} \) becomes \( 2 * \log_b{x} \), and \( \log_b(z^{2}) \) becomes \( 2 * \log_b{z} \). Now the expression is \( 2 * \log_b{x} + \log_b{y} - 2 * \log_b{z} \).

Key Concepts

Logarithmic Properties

Logarithmic properties are useful tools in simplifying logarithmic expressions. They are based on the principles and operations of logarithms. Understanding these properties can enable you to break down complex expressions into simpler terms. The basic properties include:

• The Product Rule: Allows adding logarithms with the same base.
• The Quotient Rule: Allows subtracting logarithms when dividing terms.
• The Power Rule: Alleviates handling exponents within logarithms.

Each of these rules helps transform a logarithm into a more manageable form, which is critical in solving logarithmic equations or simplifying expressions. With these properties, tricky calculations become much more straightforward!

Quotient Rule

The quotient rule is a valuable logarithmic property that allows the simplification of the logarithm of a fraction. According to this rule, the logarithm of a quotient is equivalent to the difference between the logarithms of the numerator and the denominator.
For instance, the expression \( \log_b\left(\frac{x}{y}\right) \) becomes \( \log_b{x} - \log_b{y} \). This approach can be very beneficial when attempting to simplify expressions and make them more approachable.
To apply the quotient rule, follow these steps:

• Identify the numerator and the denominator in the fraction.
• Take the logarithm of both the numerator and denominator separately.
• Subtract the logarithm of the denominator from the logarithm of the numerator.

This will leave you with a simpler expression that is easier to handle.

Product Rule

The product rule of logarithms is another key property used to simplify expressions. This rule states that the logarithm of a product is the sum of the logarithms of the factors of that product. Simply put, multiplying factors becomes addition in the realm of logarithms.
For example, in the expression \( \log_b{(xy)} \), you can apply the product rule and write it as \( \log_b{x} + \log_b{y} \). It turns multiplication in regular expressions into addition in logarithmic form.
Here's how to apply the product rule:

• Break down the product into its individual factors.
• Take the logarithm of each factor separately.
• Add together all these simpler logarithmic expressions.

Using this rule simplifies manipulation of logarithmic expressions, particularly when combined with other properties.

Power Rule

The power rule of logarithms provides a neat way to dispense with exponents in logarithmic expressions. According to this rule, the logarithm of a number raised to an exponent is the product of that exponent and the logarithm of the base. So, an exponent becomes a factor outside the logarithm.
Consider the expression \( \log_b{(x^n)} \). By applying the power rule, you simplify it to \( n \cdot \log_b{x} \). This property elegantly removes complexity by transforming exponents into coefficients.
To apply the power rule effectively:

• Identify the base and its exponent in the expression.
• Multiply the exponent with the logarithm of the base.
• Rewrite the expression in this new simplified form.

Using the power rule allows you to cleanly deal with powers without being bogged down by complicated calculations.