Problem 68 - A Graphical Approach to Precalculus with Limits

Question

Use the properties of logarithms to rewrite each logarithm if possible. Assume that all variables represent positive real numbers. $$\log _{3} \frac{4 p}{q}$$

Step-by-Step Solution

Answer

\( \log_{3} \frac{4p}{q} = \log_{3} 4 + \log_{3} p - \log_{3} q \)

Step 1: Identify the Logarithmic Property to Use

The expression given is \( \log_{3} \left( \frac{4p}{q} \right) \). To rewrite this expression, we can use the property of logarithms related to division, which states: \( \log_{b} \left( \frac{M}{N} \right) = \log_{b} M - \log_{b} N \).

Step 2: Apply the Division Property of Logarithms

Using the division property identified in Step 1, we rewrite the expression: \( \log_{3} \left( \frac{4p}{q} \right) = \log_{3} 4p - \log_{3} q \).

Step 3: Further Simplify Using the Product Property

Now, apply the logarithmic property for multiplication, which is \( \log_{b} (MN) = \log_{b} M + \log_{b} N \), to the term \( \log_{3} 4p \). This gives us: \( \log_{3} 4p = \log_{3} 4 + \log_{3} p \).

Step 4: Write the Final Expression

Substituting the result from Step 3 back into the expression from Step 2, we have: \( \log_{3} 4p - \log_{3} q = (\log_{3} 4 + \log_{3} p) - \log_{3} q = \log_{3} 4 + \log_{3} p - \log_{3} q \).

Step 5: Final Simplified Expression

Combine all the terms found: \( \log_{3} \frac{4p}{q} = \log_{3} 4 + \log_{3} p - \log_{3} q \).

Key Concepts

Division Property of Logarithms

Let's begin by exploring the division property of logarithms. This property is extremely useful when dealing with expressions that involve division inside a logarithm. What it tells us is quite simple but powerful: the logarithm of a quotient is the difference of the logarithms of the numerator and the denominator.

When you see something like \( \log_{b} \left( \frac{M}{N} \right) \), the division property comes into play. It transforms this expression into \( \log_{b} M - \log_{b} N \). This equation can simplify calculations and expressions by breaking down complex ratios into more manageable parts.

• Example: If you have \( \log_{3} \left( \frac{4p}{q} \right) \), using the division property results in: \( \log_{3} 4p - \log_{3} q \).

By breaking it down into these simpler components, it often becomes easier to analyze or further manipulate the expression.

Product Property of Logarithms

Next up, let's delve into the product property of logarithms, which helps to handle multiplication inside logarithms. According to this property, the logarithm of a product can be expressed as the sum of the logarithms of the factors.

In mathematical form, for an expression like \( \log_{b} (MN) \), it transforms into \( \log_{b} M + \log_{b} N \). By converting multiplication into an addition of separate logarithms, it can be easier to simplify or evaluate logarithmic expressions.

• Example: In our case, \( \log_{3} 4p \) can be rewritten using the product property as: \( \log_{3} 4 + \log_{3} p \).

This product property therefore allows us to take an often complex expression and make it broken down into more approachable parts.

Simplification of Logarithmic Expressions

Finally, let's talk about the simplification of logarithmic expressions. Using the properties discussed, we can streamline complex logarithmic expressions into their simplest forms.

The key is to incrementally apply the properties, breaking down the expression one piece at a time. By sequentially using both the division and product properties, you can simplify to a form that's straightforward to interpret.

In our example \( \log_{3} \frac{4p}{q} \), we first applied the division property to get \( \log_{3} 4p - \log_{3} q \). Then, we used the product property on \( \log_{3} 4p \), transforming it into \( \log_{3} 4 + \log_{3} p \).

Combining these operations, we reach a final simplified form: \( \log_{3} 4 + \log_{3} p - \log_{3} q \).

The process of simplification is not just about reaching a final answer. It aids in understanding the relationships between different parts of the expression, making these kinds of mathematical puzzles easier and more intuitive to solve.