Problem 37 - Precalculus Mathematics for Calculus

Question

Use the Laws of Logarithms to expand the expression. $$\ln (x \sqrt{\frac{y}{z}})$$

Step-by-Step Solution

Answer

\( \ln (x \sqrt{\frac{y}{z}}) = \ln x + \frac{1}{2} \ln y - \frac{1}{2} \ln z \)

Step 1: Apply the Product Rule

The expression is a natural logarithm of a product. Use the product rule of logarithms, which states that \( \ln(a \cdot b) = \ln a + \ln b \), to separate the terms inside the logarithm. For the given expression, this means: \[ \ln (x \sqrt{\frac{y}{z}}) = \ln x + \ln \left( \sqrt{\frac{y}{z}} \right) \]

Step 2: Simplify the Square Root

The expression inside the logarithm contains a square root. Recall that \( \sqrt{a} = a^{1/2} \). Therefore, rewrite \( \sqrt{\frac{y}{z}} \) as \( \left( \frac{y}{z} \right)^{1/2} \) and substitute this into the equation:\[ \ln \left( \sqrt{\frac{y}{z}} \right) = \ln \left( \left( \frac{y}{z} \right)^{1/2} \right) \] Apply this back into the equation from Step 1:\[ \ln (x \sqrt{\frac{y}{z}}) = \ln x + \ln \left( \left( \frac{y}{z} \right)^{1/2} \right) \]

Step 3: Apply the Power Rule

Use the power rule of logarithms, \( \ln(a^b) = b \ln a \), to bring down the exponent in the expression.\[ \ln \left( \left( \frac{y}{z} \right)^{1/2} \right) = \frac{1}{2} \ln \left( \frac{y}{z} \right) \]Substitute back into the equation:\[ \ln (x \sqrt{\frac{y}{z}}) = \ln x + \frac{1}{2} \ln \left( \frac{y}{z} \right) \]

Step 4: Apply the Quotient Rule

Finally, apply the quotient rule which states that \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \) to the expression:\[ \frac{1}{2} \ln \left( \frac{y}{z} \right) = \frac{1}{2} (\ln y - \ln z) \]Incorporating this back to the overall equation gives:\[ \ln (x \sqrt{\frac{y}{z}}) = \ln x + \frac{1}{2} \ln y - \frac{1}{2} \ln z \]

Step 5: Final Expression

By combining all the steps together, we have expanded the original expression using the properties of logarithms:\[ \ln (x \sqrt{\frac{y}{z}}) = \ln x + \frac{1}{2} \ln y - \frac{1}{2} \ln z \]

Key Concepts

Product Rule of Logarithms

Logarithms are powerful tools in mathematics that allow you to manage multiplication in a unique way. The product rule of logarithms is one essential method to simplify expressions involving products. According to the product rule, if you are taking the logarithm of a product of two numbers, you can separate the logarithms, like so:

• Given: \( \ln(a \cdot b) \)
• Rule: \( \ln(a \cdot b) = \ln a + \ln b \)

For example, suppose you have an expression like \( \ln(x \sqrt{\frac{y}{z}}) \). You can apply the product rule to split the expression inside the logarithm into:

• \( \ln x + \ln \left( \sqrt{\frac{y}{z}} \right) \)

This transformation makes further simplifications much clearer and easier to handle. Breaking the expression apart using the product rule sets the stage for applying other logarithmic rules, like the power and quotient rules.

Power Rule of Logarithms

The power rule of logarithms simplifies the exponent inside a logarithmic expression by allowing you to bring the exponent out in front of the logarithm. This rule is particularly useful when you deal with powers or roots within logarithmic expressions. The power rule states that:

• Given: \( \ln(a^b) \)
• Rule: \( \ln(a^b) = b \ln a \)

In the expression \( \ln \left( \left( \frac{y}{z} \right)^{1/2} \right) \), you can use the power rule by identifying the exponent \( \frac{1}{2} \). The rule helps break down this complex expression into:

• \( \frac{1}{2} \ln \left( \frac{y}{z} \right) \)

This simplification makes it easier to further expand the expression using other rules, such as the quotient rule, next. By moving the exponent outside of the logarithm, calculations become more straightforward.

Quotient Rule of Logarithms

The quotient rule of logarithms is beneficial when dealing with division inside a logarithmic function. The rule allows you to split a logarithm of a quotient into a difference of two logarithms. It is expressed as:

• Given: \( \ln \left( \frac{a}{b} \right) \)
• Rule: \( \ln \left( \frac{a}{b} \right) = \ln a - \ln b \)

In the context of the expression \( \frac{1}{2} \ln \left( \frac{y}{z} \right) \), the quotient rule assists in breaking it down further to:

• \( \frac{1}{2} (\ln y - \ln z) \)

This breakdown not only makes it easier to understand the components of the expression but also allows for further simplification and clarity. By using the quotient rule, the interplay of division within the logarithm is unraveled into more manageable parts.