Problem 21 - Precalculus : Building Concepts and Connections

Question

In Exercises \(21-30,\) write each logarithm as a sum and\or difference of logarithmic expressions. Eliminate exponents and radicals and evaluate logarithms wherever possible. Assume that \(a, x, y\) \(z>0\) and \(a \neq 1\). $$\log \frac{x^{2} y^{5}}{10}$$

Step-by-Step Solution

Answer

The solution is \( 2\log(x) + 5\log(y) - 1 \).

Step 1: Figure out the fraction within the logarithm

Find the fraction inside the logarithm which is \( \frac{x^{2} y^{5}}{10} \).

Step 2: Apply the quotient rule of logarithms

The quotient rule states that \( \log_{a}(\frac{m}{n}) = \log_{a}m - \log_{a}n \). Apply this rule to our fraction: \( \log(x^{2}y^{5}) - \log(10) \).

Step 3: Break Up the Products

Use the product rule which states that \( \log_{a}mn = \log_{a}m + \log_{a}n \) to break up the products: \( \log(x^{2}) + \log(y^{5}) - \log(10) \).

Step 4: Apply the power rule of logarithms

The power rule states that \( \log_{a}m^{n} = n \log_{a}m \). Apply this rule to remove the exponents: \( 2\log(x) + 5\log(y) - \log(10) \). Remember that \( \log(10) \) is simply 1.

Step 5: Final simplification

Simplify the expression to the final form: \( 2\log(x) + 5\log(y) - 1 \).

Key Concepts

Quotient Rule of Logarithms

The quotient rule of logarithms is an essential concept for simplifying logarithmic expressions. When you have a logarithm of a fraction, such as \( \log_a \left( \frac{m}{n} \right) \), the quotient rule allows you to separate the expression into a difference of two logarithms. Specifically, it tells us that:

• \( \log_a \left( \frac{m}{n} \right) = \log_a m - \log_a n \)

This rule is particularly useful because it transforms division within a logarithm into a simple subtraction operation outside the logarithm.
For example, to apply the quotient rule to \( \log \left( \frac{x^2 y^5}{10} \right) \), you first identify the numerator and denominator. Then, apply the rule to write:\( \log(x^2 y^5) - \log(10) \).
This approach makes the expression easier to handle, setting the stage for further simplification using other rules.

Product Rule of Logarithms

The product rule of logarithms helps in dealing with logarithms that contain products inside. According to the product rule, when you have a product within a logarithm, it can be split into a sum of two logarithms:

• \( \log_a(mn) = \log_a m + \log_a n \)

This feature is valuable because it breaks the problem down into smaller parts, allowing you to tackle each part individually.
In our initial expression, \( \log(x^2 y^5) \), applying the product rule gives us:\( \log(x^2) + \log(y^5) \).
By turning a product into a sum of logs, it becomes simpler to manage each logarithmic term independently, paving the way for using additional rules like the power rule.

Power Rule of Logarithms

Once you've separated the logarithmic expression using the quotient and product rules, the power rule helps eliminate exponents from the logarithmic statements. This rule dictates that when you have a power inside a logarithm, you can bring the exponent in front of the logarithm:

• \( \log_a(m^n) = n\log_a m \)

The power rule is particularly helpful for simplifying expressions by reducing the complexity of exponents within logarithmic terms.
In our example, after breaking down the expression, we have \( \log(x^2) + \log(y^5) \).
Using the power rule, these terms convert into \( 2\log(x) + 5\log(y) \), making them easier to evaluate and understand. By doing so, the logarithmic expression becomes a simple addition, represented by known logarithmic values.