Problem 68 - 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.)\(\log _{6} 6 x\)

Step-by-Step Solution

Answer

The expansion of the expression \(\log_6{6x}\) as a sum, difference, and/or multiple of logarithms is \(1 + \log_6{x}\)

Step 1: Identify the Logarithm Base

In this exercise, the base of the logarithm is 6. It should be noted that logarithms with the same base can be combined or split using the properties of logarithms.

Step 2: Apply the first property of Logarithms

The expression can be expanded as a sum of logarithms using the first logarithm property (\(\log_b(a*b) = \log_b(a) + \log_b(b)\)). In this case, 6 and x are being multiplied inside the logarithm. Therefore, \(\log_6{6x}\) can be rewritten as \(\log_6{6} + \log_6{x}\).

Step 3: Simplify the Logarithm

Simplify the expression \(\log_6{6}\). Because any value with a logarithm that matches its base equals 1, \(\log_6{6}\) can be simplified to 1. This leaves the expanded expression as \(1 + \log_6{x}\).

Key Concepts

Logarithm Properties

Understanding the properties of logarithms is key to simplifying complex logarithmic expressions and solving logarithmic equations. Essentially, logarithms are exponents. The logarithm of a number is the exponent to which the base must be raised to produce that number. For example, consider the logarithmic equation \( \log_b(a) = c \). This equation is saying that the base \( b \) raised to the power of \( c \) equals \( a \).

There are several properties that are particularly useful: the product rule (\( \log_b(mn) = \log_b(m) + \log_b(n) \) for any positive numbers \( m \) and \( n \) and base \( b \) greater than 0), the quotient rule (\( \log_b\left(\frac{m}{n}\right) = \log_b(m) - \log_b(n) \) where \( m \) and \( n \) are positive numbers), and the power rule (\( \log_b(m^c) = c\cdot\log_b(m) \) for any positive number \( m \) and any real number \( c \)). Understanding how to apply these properties allows us to rewrite and simplify logarithmic expressions strategically.

Logarithm Simplification

The process of breaking down a complex logarithmic expression into simpler parts often requires using the logarithm properties we've just described. When we simplify logarithms, we aim to make them more comprehensible and manageable, particularly when solving equations. For example, the rule that \( \log_b(b) = 1 \) is a fundamental simplification property because it indicates that the logarithm of a number to its own base is always 1. This is why in our given exercise, \( \log_6{6} \) simplifies to 1.

Simplification can turn multiplication into addition, division into subtraction, and exponents into coefficients. Continuing with our exercise, which has now been simplified to \( 1 + \log_6{x} \), makes the expression much easier to understand and use in calculations. By mastering simplification techniques, students can handle complex logarithmic operations with confidence.

Logarithm Base

The base of a logarithm is a crucial component, as it dictates how the logarithm behaves. In the given exercise, the base is 6, indicated by the subscript on the log symbol (\( \log_6 \)). The base tells us what number we are using as the 'ground level' for our exponential operations. When we change the base, we change the entire scale of the logarithmic function.

It's essential to know how to handle different bases. For logarithms with the same base, as we've seen in the exercise, we can use the properties of logarithms to work with and simplify expressions. However, when dealing with logarithms of different bases, we may need to employ specialty rules such as the change of base formula, which allows us to convert a logarithm to any other base. Understanding bases enables students to navigate between different logarithmic scales, making it easier to compare and solve logarithmic expressions.