Problem 258 - Precalculus

Question

For the following exercises, condense to a single logarithm if possible. $$ \log _{3}(2)+\log _{3}(a)+\log _{3}(11)+\log _{3}(b) $$

Step-by-Step Solution

Answer

The expression condenses to \(\log_3(22ab)\).

Step 1: Identify Logarithmic Property

The expression involves a sum of logarithms. Recognize that the sum of logs with the same base can be expressed as the logarithm of a product: \(log_b(x) + log_b(y) = log_b(x \cdot y)\). This property is used to condense the expression into a single logarithm.

Step 2: Apply Property to Pair of Logarithms

Starting with the first two terms: \(log_3(2) + log_3(a)\), apply the product property to get \(log_3(2 \cdot a)\).

Step 3: Combine Next Logarithm

Now combine the result from the second step with \(log_3(11)\). Use the product property again: \(log_3((2 \cdot a) \cdot 11) = log_3(22a)\).

Step 4: Combine Final Logarithm

Use the product property a final time to include \(log_3(b)\) with \(log_3(22a)\). This gives: \(log_3((22a) \cdot b) = log_3(22ab)\).

Step 5: Final Expression

The final condensed expression is \(log_3(22ab)\). This represents the original sum of logarithms condensed into a single logarithm.

Key Concepts

Condensing Logarithms

Condensing logarithms is a process where multiple logarithms are combined into a single expression. This is particularly useful in simplifying logarithmic expressions or solving logarithmic equations. The main tool we use for this process is the logarithm properties. By recognizing patterns or structures of the logarithms and applying the appropriate properties, we can rewrite the initial complex expressions in a more compact form.

Here's why condensing is valuable:

• It simplifies expressions, making them easier to calculate or interpret.
• It allows you to clearly see relationships or functions that might be hidden in longer equations.
• It prepares expressions for further algebraic manipulation or integration into broader problem-solving methods.

In our exercise, we applied the product property of logarithms to condense four separate logarithms into one expression. This condensed form is generally more elegant and convenient to work with.

Product Property of Logarithms

The product property of logarithms is a fundamental rule that states if you have a sum of two log expressions with the same base, you can combine them into a single log expression of a product. Mathematically, it’s expressed as: \[\log_b(x) + \log_b(y) = \log_b(x \cdot y)\]

Understanding and applying this property helps in condensing expressions by combining logarithms into a single term. In our example, this property was used multiple times across several steps:

• First, used to condense \(\log_3(2) + \log_3(a)\) into \(\log_3(2a)\).
• Then, applied again to combine \(\log_3(2a) + \log_3(11)\), transforming it into \(\log_3(22a)\).
• Finally, another application to incorporate \(\log_3(b)\), yielding \(\log_3(22ab)\).

By mastering this property, you can considerably simplify complex logarithmic expressions, making them more manageable and concise.

Sum of Logarithms

When dealing with the sum of logarithms, it's crucial to recognize that you can often simplify these by employing the properties of logarithms. The expression of a sum in log form, like the one in our exercise, provides the perfect opportunity to practice condensing. For the given expression:\[\log _{3}(2)+\log _{3}(a)+\log _{3}(11)+\log _{3}(b)\]

Each logarithm in the sum shares the same base. This is key when simplifying because the product property of logarithms only applies under this condition. By sequentially applying the property to pairs of terms, each step reduces the number of logarithms until you achieve the condensed form.

• This technique streamlines calculations where the full expansion of products would otherwise be cumbersome.
• It reinforces understanding of how products and powers relate in logarithms, widening your grasp on manipulating terms algebraically.

Thus, recognizing and working through the sum of logarithms harnesses the power of these properties to reach more easily solvable forms.

Precalculus

Precalculus serves as the bridge between algebra, geometry, and calculus, incorporating essential concepts that underpin more advanced mathematical topics. Logarithms, and specifically the manipulation of logarithmic expressions, play a crucial role in precalculus, preparing students for calculus challenges.

Key aspects of precalculus related to our current topic include:

• Understanding properties of logarithms, such as product, quotient, and power rules.
• Equipping students with skills to manage and transform expressions for solving equations.
• Incorporating graphical interpretations of logarithmic and exponential functions, providing a visual context.

By mastering these foundational skills within precalculus, students build robust analytical skills necessary for tackling calculus problems. You’ll find that these logarithmic properties are not just beneficial but essential for modeling real-world phenomena in engineering, science, and finance, making precalculus a significant step in any mathematical journey.