Problem 74 - Intermediate Algebra

Question

For Problems \(69-80\), express each of the following as the sum or difference of simpler logarithmic quantities. Assume all variables represent positive real numbers. For example, $$ \begin{aligned} \log _{b} \frac{x^{3}}{y^{2}} &=\log _{b} x^{3}-\log _{b} y^{2} \\ &=3 \log _{b} x-2 \log _{b} y \end{aligned} $$ $$ \log _{b} x^{2} y^{3} $$

Step-by-Step Solution

Answer

\(\log_{b}(x^{2}y^{3}) = 2\log_{b}(x) + 3\log_{b}(y)\)

Step 1: Identify the Expression

We need to simplify the expression \( \log_{b} (x^{2} y^{3}) \) using the properties of logarithms.

Step 2: Apply the Product Rule of Logarithms

The product rule of logarithms states that \( \log_{b}(MN) = \log_{b}M + \log_{b}N \). Apply this rule to the expression \( \log_{b} (x^{2} y^{3}) \) to separate the terms: \[\log_{b} (x^{2} y^{3}) = \log_{b} (x^{2}) + \log_{b} (y^{3}).\]

Step 3: Apply the Power Rule of Logarithms

The power rule of logarithms states that \( \log_{b}(M^{n}) = n \log_{b}M \). Apply this rule to each term in the expression: \[\log_{b} (x^{2}) + \log_{b} (y^{3}) = 2\log_{b}(x) + 3\log_{b}(y).\]

Step 4: Combine the Results

Combine the results from Step 3 to express the original logarithmic expression in terms of simpler logarithmic quantities: \[\log_{b} (x^{2} y^{3}) = 2\log_{b}(x) + 3\log_{b}(y).\]

Key Concepts

Product Rule of Logarithms

The product rule of logarithms is a helpful property that simplifies the handling of logarithms when you are multiplying two components inside the logarithm. It says that the logarithm of a product is the same as the sum of the logarithms of the individual factors. This property can be summarized with the formula:
\( \log_{b}(MN) = \log_{b}M + \log_{b}N \).
Why is this useful? Instead of dealing with a complex product within the logarithm, you can break it down into several simpler parts. Each part can be understood and handled individually. For instance, if you encounter \( \log_{b} (x^{2} y^{3}) \), using the product rule allows you to interpret it as two separate logarithms:

• \( \log_{b} x^{2} \)
• \( \log_{b} y^{3} \)

When learning logarithms, this rule is instrumental in making complicated expressions more manageable by changing a multiplication task into an addition task. This generally aligns better with basic arithmetic skills and makes further manipulation and simplification much more straightforward.

Power Rule of Logarithms

The power rule of logarithms is another critical principle that makes handling logarithmic expressions simpler. This rule comes into play when we have an exponent within the logarithm. It states that the logarithm of a power can be calculated by multiplying the exponent by the logarithm of the base. Mathematically, this translates to:
\( \log_{b}(M^{n}) = n \log_{b}M \).
Using this rule, logarithmic components with exponents shrink into manageable pieces. Consider the expression \( \log_{b} x^{2} \): the power rule tells us this is equivalent to \( 2 \log_{b} x \).

• \( \log_{b} (x^{2}) = 2\log_{b}(x) \)
• Similarly, \( \log_{b} (y^{3}) = 3\log_{b}(y) \)

Learning and applying this principle makes dealing with exponents inside logarithms significantly more manageable. By multiplying the exponent with the logarithm of the base, you convert a problem involving exponents into a simpler multiplication issue.

Simplifying Logarithms

Simplifying logarithmic expressions means breaking them down into straightforward, basic components. The goal is to express complex logarithms as sums or differences of simpler logarithmic terms. This involves using rules like the product rule and power rule.
Let’s go through a practical example. If you start with a logarithmic expression like \( \log_{b} (x^{2} y^{3}) \), you commence by applying the product rule to separate \( x^{2} \) and \( y^{3} \):

• Break it down: \( \log_{b} (x^{2}) + \log_{b} (y^{3}) \)

After using the product rule, you use the power rule on each term:

• For \( \log_{b}(x^{2}) \), it becomes \( 2\log_{b}(x) \)
• For \( \log_{b}(y^{3}) \), it becomes \( 3\log_{b}(y) \)

This simplification results in:

• Final form: \( 2\log_{b}(x) + 3\log_{b}(y) \)

By employing these rules, complicated expressions become sums (or differences) of simpler logarithmic quantities, enhancing understanding and ease of calculation.