Problem 27 - College Algebra

Question

In Exercises \(1-40,\) 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 the expression \( \log_b{\left(\frac{x^{2} y}{z^{2}}\right)} \) is \(2 \log_b{(x)} + \log_b{(y)} - 2 \log_b{(z)}\).

Step 1: Apply Quotient Rule

Begin by applying the quotient rule of logarithms. This rule states that the logarithm of a quotient is equal to the difference of the logarithm of the numerator and the logarithm of the denominator. Therefore, the expression can be re-written as follows: \( \log_b{(x^2y)} - \log_b{(z^2)} \).

Step 2: Apply Product Rule

Next, apply the product rule. The product rule for logarithms states that the log of a product is equal to the sum of the logs of its factors. Apply the rule to the numerator part of our expression to obtain \( \log_b{(x^2)} + \log_b{(y)} - \log_b{(z^2)} \).

Step 3: Apply Power Rule

And finally, apply the power rule. The power rule of logarithms states that the log of a number raised to an exponent is equal to the product of the log of that number and the exponent. Utilize this rule to the term with the power to achieve the expanded form of the given expression, which is: \(2\log_b{(x)} + \log_b{(y)} - 2\log_b{(z)} \).

Key Concepts

Quotient Rule of Logarithms

Understanding the Quotient Rule is key when working with logarithms. This rule states that the logarithm of a quotient is the difference between the logarithm of the numerator and the logarithm of the denominator. It simplifies expressions that involve division by breaking them down into subtraction.

For instance, if you have a logarithmic expression like \( \log_b \left( \frac{x}{y} \right) \), you can apply the quotient rule. This converts the expression into \( \log_b(x) - \log_b(y) \). By doing so, it becomes easier to further manipulate or evaluate, especially in complex equations with multiple variables. Always remember, the base \( b \) in the logs involved remains the same.

Keep this concept in mind:

• The base of the logarithm must be identical for the rule to apply effectively.
• The answer you get is dependent on subtracting the logarithm of the divisor from the dividend.

Product Rule of Logarithms

The Product Rule for logarithms is another fundamental law that assists in simplifying expressions. According to this rule, the logarithm of a product is the sum of the logarithms of its individual factors. This law helps when dealing with expressions that multiply several terms under a single log function.

Imagine breaking down \( \log_b(xy) \) into its components. By utilizing the product rule, it becomes \( \log_b(x) + \log_b(y) \). This is particularly useful because it transforms multiplication inside the logarithm into addition outside. It is easier to work through subsequent calculations involving these separate logs.

Things to keep in mind:

• The product rule applies only when factors are under a single logarithm.
• The base \( b \) of the logarithm remains unchanged during transformation.

Power Rule of Logarithms

The Power Rule is yet another important property of logarithms, especially useful when dealing with expressions involving exponents. This rule states that for a logarithm of a term raised to a power, the exponent can be taken out as a multiplier. This aids in simplifying expressions significantly.

For example, if you see \( \log_b(x^n) \), using the power rule, transform it to \( n\log_b(x) \). Essentially, instead of working with the exponent inside the logarithm, we multiply this exponent by the logarithm of the base number. It's a powerful tool to make the expressions less cumbersome and more manageable.

Essential points:

• Only apply the power rule when the entire term is raised to a power, not just parts of it.
• The base \( b \) in the logarithmic expression does not change.