Problem 46 - Precalculus with Limits

Question

Expanding a Logarithmic Expression In Exercises \(37-58,\) use the properties of logarithms to expand the expression as a sum, difference, and or constant multiple of logarithms. (Assume all variables are positive.) $$\log 4 x^{2} y$$

Step-by-Step Solution

Answer

The expanded form of the logarithmic expression \(\log 4 x^{2} y\) is \(\log 4 + 2 \log x + \log y\) .

Step 1: Identify the Properties of Logarithms to be used

Look at the given logarithmic expression \(\log 4 x^{2} y\). It is a product of terms, 4, \(x^2\) and \(y\). This relates to the logarithmic property \(\log(a*b) = \log(a) + \log(b)\), here \(a=4\), \(b=x^2 y\).

Step 2: Apply the Logarithmic Product Property

Apply the product rule of logarithms to split the logarithms. The expression \(\log 4 x^{2} y\) becomes \(\log 4 + \log x^{2} y\).

Step 3: Apply the Logarithmic Product Property Again

Looking at the second term \(\log x^{2} y\), it also represents the product of \(x^2\) and \(y\), apply the product rule of logarithms again to split it. The expression \(\log 4 + \log x^{2} y\) becomes \(\log 4 + \log x^{2} + \log y\).

Step 4: Apply the Power Property of Logarithm

Looking at the second term \( \log x^{2}\), use the power property of logarithms which states log(a^n) = n*log(a). So, \( \log x^{2} \) becomes 2*log(x). So, the final expression is \(\log 4 + 2 \log x + \log y\).

Key Concepts

Properties of Logarithms

Understanding the properties of logarithms is essential for expanding logarithmic expressions effectively. Logarithms have distinct properties that allow us to transform complex expressions into simpler components that are easier to work with.

Firstly, logarithms convert multiplication into addition - this is a fundamental concept known as the logarithmic product property. Another core property is the power property, which allows us to handle exponents within logarithmic functions. Moreover, logarithms can transform division into subtraction, although this property isn't required for the current exercise.

These properties have practical applications in solving exponential equations, analyzing growth patterns, and simplifying complex expressions. Mastering them will give students a valuable toolset for dealing with logarithmic functions across various mathematical problems.

Logarithmic Product Property

The logarithmic product property is a powerful tool for breaking down expressions involving products under a single logarithm. Simply put, this property states that the logarithm of a product is equal to the sum of the logarithms of the factors.
For example, the property can be expressed as: \[\begin{equation} \log(ab) = \log(a) + \log(b) \end{equation}\]
In the exercise, we see this property in action as the expression \(\log 4x^{2}y\) is simplified to \(\log 4 + \log x^{2} + \log y\). It enables the separation of terms so they may be handled individually, making complex problems more manageable.

Power Property of Logarithms

The power property of logarithms is another useful property that gives us a method to deal with exponents inside logarithms. This property says that the logarithm of a power is equal to the exponent times the logarithm of the base. Formally, it's written as:\[\begin{equation} \log(a^n) = n \cdot \log(a) \end{equation}\]
This comes into play when simplifying the term \(\log x^{2}\) in our exercise. By applying the power property, it simplifies to \(2\log x\), streamlining the expression further. Grasping this concept helps in both expanding and condensing logarithmic expressions, proving invaluable when dealing with growth and decay problems, or even in financial calculations involving compound interest.