Problem 65 - College Algebra and Calculus: An Applied Approach

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or multiple of logarithms. (Assume all variables are positive.)\(\log _{2}\left(4^{3} \cdot 3^{5}\right)\)

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(6 + 5\log_{2}{3}\)

Step 1: Apply the product rule

The first step is to apply the product rule of logarithms: \(\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)\).\nUsing this rule, the given expression \(\log _{2}\left(4^{3} \cdot 3^{5}\right)\) can be expanded to: \(\log_{2}(4^{3}) + \log_{2}(3^{5})\)

Step 2: Apply the power rule

Next, apply the power rule of logarithms: \(\log_{a}(b^c) = c\log_{a}(b)\).\nFollowing that rule, the expression becomes: \(3\log_{2}(4) + 5\log_{2}(3)\)

Step 3: Apply the base change rule

For \(\log_{2}(4)\), we can further simplify it using the base change rule since 4 is a power of 2. The base change rule is defined as: \(\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\)\nThis simplifies \(\log_{2}(4)\) to 2 since 4 is equal to \(2^{2}\). Thus, the final expanded logarithmic expression is: \(3*2 + 5\log_{2}(3)\)

Key Concepts

Understanding the Product Rule of Logarithms

In logarithmic calculus, the product rule is a vital fundamental property. It allows us to split the logarithm of a multiplied product into the sum of the logarithms of individual factors. This is immensely helpful in simplifying expressions and is represented as:

• \(\log_{a}(bc) = \log_{a}(b) + \log_{a}(c)\)

Using the product rule, we can transform complex logarithmic expressions into easier-to-understand terms. For instance, in the example \(\log_{2}(4^{3} \cdot 3^{5})\), each component of the multiplication \(4^{3}\) and \(3^{5}\) is treated separately in the expression \(\log_{2}(4^{3}) + \log_{2}(3^{5})\).
This effectively reduces the computational complexity as we deal with simpler logs.
Mastery of this rule ensures accurate manipulation of multiplicative factors within logarithmic functions.

Applying the Power Rule of Logarithms

The power rule of logarithms provides us a method to deal with exponents within logarithmic expressions. By understanding this rule, you can transform the logarithm of a power into a multiplication involving the exponent itself. This rule is expressed as:

• \(\log_{a}(b^c) = c \cdot \log_{a}(b)\)

In using it, the exponent \(c\) becomes a coefficient in front of the log.
Let's see it in action with our example: when facing \(\log_{2}(4^{3}) + \log_{2}(3^{5})\), you use the power rule to simplify it to \(3 \cdot \log_{2}(4) + 5 \cdot \log_{2}(3)\).
Doing so not only simplifies the expression but also makes it computationally friendly.
The power rule is vital for handling large exponents efficiently within logarithmic contexts.

Utilizing the Base Change Rule of Logarithms

The base change rule is an advanced tool in the logarithm toolbox that allows for the transformation of logarithmic expressions between different bases. It is particularly useful when dealing with non-trivial bases or when simplifying expressions to their simplest form. This rule is written as:

• \(\log_{b}(a) = \frac{\log_{c}(a)}{\log_{c}(b)}\)

In practice, the base change rule can make complex calculations much simpler. For example, in the exercise: after we've simplified to \(3\log_{2}(4) + 5\log_{2}(3)\), you can simplify \(\log_{2}(4)\) further by using this rule, noting that \(4 = 2^2\).
Thus, \(\log_{2}(4) = \frac{\log_{2}(4)}{\log_{2}(2)} = 2\).
Therefore, the expression reduces to \(3 \times 2 + 5 \log_{2}(3)\).
The base change rule thus streamlines calculations especially when dealing with powers of the same base.