Problem 18 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the expression as the logarithm of a single quantity. $$\frac{1}{2} \ln x+2 \ln y-3 \ln z$$

Step-by-Step Solution

Answer

The given expression can be simplified using logarithmic properties as follows: \(\frac{1}{2} \ln x + 2 \ln y - 3 \ln z = \ln\left(\frac{x^{\frac{1}{2}} \cdot y^2}{z^3}\right)\)

Step 1: Identify the logarithmic properties to simplify the expression

Here we will use the following logarithm properties: 1. \(log_b(m \cdot n) = log_b(m) + log_b(n)\) 2. \(log_b\left(\frac{m}{n}\right) = log_b(m) − log_b(n)\) 3. \(log_b\left({m}^{n}\right) = n \cdot log_b(m)\) These properties will be applied to the given expression to make it a logarithm of a single quantity.

Step 2: Apply the power law

We can start by applying property 3 to the coefficients of the logarithms: \(\frac{1}{2} \ln x + 2 \ln y - 3 \ln z = \ln\left(x^{\frac{1}{2}}\right) + \ln\left(y^2\right) - \ln\left(z^3\right)\)

Step 3: Combine the logarithms using properties

Now, we will apply the properties 1 and 2 to the updated expression: \(\ln\left(x^{\frac{1}{2}}\right) + \ln\left(y^2\right) - \ln\left(z^3\right) = \ln\left(\frac{x^{\frac{1}{2}} \cdot y^2}{z^3}\right)\)

Step 4: Write the final expression

We now have the expression written as a single logarithm: \(\frac{1}{2} \ln x + 2 \ln y - 3 \ln z = \ln\left(\frac{x^{\frac{1}{2}} \cdot y^2}{z^3}\right)\)

Key Concepts

Logarithm Properties

Logarithms have several key properties that help simplify complex expressions. These properties make it easier to manipulate and solve expressions involving logarithms.
Here are the main properties of logarithms you should remember:

• Product Rule: \( \log_b(m \cdot n) = \log_b(m) + \log_b(n) \) - This property states that the logarithm of a product is equal to the sum of the logarithms of the factors.
• Quotient Rule: \( \log_b\left( \frac{m}{n} \right) = \log_b(m) - \log_b(n) \) - This property explains that the logarithm of a quotient is the difference between the logarithms of the numerator and the denominator.
• Power Rule: \( \log_b(m^n) = n \cdot \log_b(m) \) - According to this rule, the logarithm of a power is the exponent times the logarithm of the base.

These properties are crucial when manipulating logarithmic expressions in both mathematical and real-world applications. They form the foundation for virtually all operations involving logarithms.

Power Law of Logarithms

One of the most powerful tools when working with logarithmic expressions is the power law. It is particularly useful when coefficients exist before the logarithm. The power law states that \( \log_b(m^n) = n \cdot \log_b(m) \).
This rule allows us to move any coefficient in front of a logarithm to an exponent on the base. For instance, when given an expression like \( \frac{1}{2} \ln x \), you can rewrite it as \( \ln(x^{\frac{1}{2}}) \).
Similarly, a term like \( 2 \ln y \) becomes \( \ln(y^2) \). This transformation often simplifies the process of combining multiple terms into a single logarithm.
By applying the power law, each log term can be adjusted to facilitate further simplification using other properties. Mastery of this law helps deal with more complex logarithmic expressions efficiently.

Simplification of Expressions

When faced with expressions involving logarithms, simplification usually involves combining multiple logs into a single expression. This process utilizes the product, quotient, and power rules of logarithms.
Consider the expression \( \frac{1}{2} \ln x + 2 \ln y - 3 \ln z \). To simplify it as a single logarithm, we apply the power rule to adjust the exponents first:

• \( \frac{1}{2} \ln x \) becomes \( \ln(x^{\frac{1}{2}}) \)
• \( 2 \ln y \) becomes \( \ln(y^2) \)
• \( -3 \ln z \) becomes \( \ln(z^{-3}) \)

Now, using the product and quotient rules, we combine these into a single expression:

• Addition of logs: \( \ln(x^{\frac{1}{2}}) + \ln(y^2) = \ln(x^{\frac{1}{2}} \cdot y^2) \)
• Subtraction of logs: \( \ln(x^{\frac{1}{2}} \cdot y^2) - \ln(z^3) = \ln\left(\frac{x^{\frac{1}{2}} \cdot y^2}{z^3}\right) \)

This meticulous approach ensures you arrive at a concise and accurate expression, thus making calculations more straightforward and easier to interpret.