Problem 72 - Precalculus with Limits

Question

Condensing a Logarithmic Expression In Exercises \(67-82\) , condense the expression to the logarithm of a single quantity. $$-4 \log _{6} 2 x$$

Step-by-Step Solution

Answer

\(-4 \log _{6} 2x\) condensed to the logarithm of a single quantity is \(\log_{6} (1/16x^4)\)

Step 1: Identify all coefficients in front of the logarithmic function

We first identify all coefficients or numbers that compare with the logarithmic function. In this case, it is -4.

Step 2: Use the power rule of logarithms

According to the power rule, we can transform \(a \log_b(c)\) into \(\log_b(c^a)\). Hence, we can express \(-4 \log_{6} 2x\) as \(\log_{6} {(2x)}^{-4}\). So, \(-4 \log _{6} 2x = \log_{6} {(2x)}^{-4}\)

Step 3: Simplify by evaluating the power

By evaluating the power \({(2x)}^{-4}\), we get the resulting expression as \(\log_{6} (1/{(2x)^4})\) which simplifies to \(\log_{6} (1/16x^4)\) which is the logarithm of a single quantity as required.

Key Concepts

Understanding the Power Rule of Logarithms

When condensing logarithmic expressions, one handy tool is the power rule of logarithms. This rule states that the logarithm of a number raised to a power is equal to the power multiplied by the logarithm of the number. In mathematical terms, this is expressed as:
\[ \log_b(c^a) = a \cdot \log_b(c) \]
This transformation is crucial for simplifying expressions and solving logarithmic equations. Let's consider our example of \(-4 \log_{6} 2x\). Applying the power rule, we turn this expression into a single logarithmic term:\[\log_{6} {(2x)}^{-4}\]. The move of -4 in front of the logarithm to the exponent position inside the logarithmic function is a textbook illustration of the power rule in action.
Remember, the base of the logarithm (in this case, 6), remains unchanged during this process. It's an essential step in condensing the expression,leaving you with a neater and often more manageable form to work with.

Exploring Logarithmic Functions

A logarithmic function is the inverse of exponentiation. You likely already know that \(b^y = x\) can be rewritten as \(y = \log_b(x)\), where '\(b\)' is the base of the logarithm. Understanding logarithmic functions is critical when working with exponential forms and finding unknown exponents.
In our previous example, \(\log_{6} (1/16x^4)\) represents a logarithmic function where 6 is the base, and \(1/16x^4\) is the argument. The base indicates the number that is raised to a power to achieve the argument. Think of the logarithm as asking the question: to what exponent must we raise 6 to get \(1/16x^4\)? It's these types of functions that we often need to simplify or condense, to reveal the underlying relationships and make calculations more straightforward.

Simplifying Logarithms

The process of simplifying logarithms is a bit like cleaning up a room; it's about making things less cluttered and more functional. Simplifying might involve combining logarithms using the properties of logarithms, or it can be as simple as applying the power rule, as we have seen.
In our given problem, once we have applied the power rule, we proceed to simplify the resulting expression. The numerator 1 remains the same, and we evaluate the negative exponent to flip our \(2x\) term to the denominator and raise it to the fourth power, which results in \(\log_{6} (1/16x^4)\).

• Remember, negative exponents translate to fractions!
• Always check if you can further simplify the expression inside the logarithm.
• The goal is to express your logarithmic term as cleanly as possible, ideally ending up with a single logarithmic expression as we did in the end with \(\log_{6} (1/16x^4)\).

Simplifying logarithms not only helps in making the problems easier to solve but also aids in understanding the nature and behavior of logarithmic relationships.