Problem 15 - College Algebra

Question

For the following exercises, use the properties of logarithms to expand each logarithm as much as possible. Rewrite each expression as a sum, difference, or product of logs. $$ \log \left(\frac{x^{15} y^{13}}{z^{19}}\right) $$

Step-by-Step Solution

Answer

The expanded form is \(15\log(x) + 13\log(y) - 19\log(z)\).

Step 1: Apply the Quotient Rule

The first step in expanding the logarithmic expression is to apply the Quotient Rule for logarithms. This rule states that \(\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\). Applying this rule to \(\log\left(\frac{x^{15}y^{13}}{z^{19}}\right)\), we get:\[\log\left(x^{15}y^{13}\right) - \log(z^{19})\]

Step 2: Apply the Product Rule

Next, we use the Product Rule which states that \(\log(ab) = \log(a) + \log(b)\). Apply this to \(\log(x^{15}y^{13})\):\[\log(x^{15}y^{13}) = \log(x^{15}) + \log(y^{13})\]Thus, substituting back, we have:\[\log(x^{15}) + \log(y^{13}) - \log(z^{19})\]

Step 3: Apply the Power Rule

Finally, we apply the Power Rule for logarithms, which states \(\log(a^b) = b\log(a)\). Apply this rule to each term:- \(\log(x^{15}) = 15\log(x)\)- \(\log(y^{13}) = 13\log(y)\)- \(\log(z^{19}) = 19\log(z)\)Substitute these back into the expression from Step 2:\[15\log(x) + 13\log(y) - 19\log(z)\]

Step 4: Final Solution

After expanding the original expression using properties of logarithms, we reach the final expanded form:\[15\log(x) + 13\log(y) - 19\log(z)\]

Key Concepts

Properties of Logarithms

Logarithms are incredibly useful in simplifying expressions, solving equations, and understanding exponential relationships. To unlock their full potential, it is important to understand their core properties:

• Quotient Rule: This helps in breaking down the division of variables.
• Product Rule: This aids in decomposing the multiplication of terms.
• Power Rule: This allows for simplifying terms that have exponents.

These properties are fundamental in mathematics, enabling us to transform complicated logarithmic expressions into simpler components.
Understanding these can provide clarity and insight when analyzing mathematical problems involving logarithms.

Quotient Rule

The Quotient Rule for logarithms is a key property that allows us to transform a logarithm of a division into the difference of two separate logarithms. The formula is given by:\[\log\left(\frac{a}{b}\right) = \log(a) - \log(b)\]This rule is highly useful when dealing with expressions where one quantity is divided by another. For example, using the exercise's expression, \(\log\left(\frac{x^{15}y^{13}}{z^{19}}\right)\), applying the Quotient Rule results in:

• \(\log(x^{15}y^{13}) - \log(z^{19})\)

By using this rule, we can take a complex logarithmic function and break it down into manageable parts, making further simplification tasks much easier.

Product Rule

The Product Rule for logarithms provides a method to break down logarithms of products into the sum of individual logarithms. It is expressed mathematically as:\[\log(ab) = \log(a) + \log(b)\]This is particularly helpful when expanding expressions that involve the multiplication of variables.
In our example, apply the Product Rule to the first part of the expanded quotient expression:

• \(\log(x^{15}y^{13}) = \log(x^{15}) + \log(y^{13})\)

This transformation allows for more detailed examination and simplification by separating the terms which can then be dealt with independently.

Power Rule

The Power Rule greatly simplifies the process of dealing with exponents within logarithmic functions. Its formula is as follows:\[\log(a^b) = b\log(a)\]This rule allows for the exponent to be brought down in front, transforming exponential terms into linear factors.
Using the Power Rule in the exercise:

• \(\log(x^{15}) = 15\log(x)\)
• \(\log(y^{13}) = 13\log(y)\)
• \(\log(z^{19}) = 19\log(z)\)

Applying this rule, we break down exponential expressions into sums and differences of logs, easily readable for further mathematical manipulation.