Problem 7 - College Algebra and Calculus: An Applied Approach

Question

Write the logarithm in terms of common logarithms.\(\log _{1 / 5} x\)

Step-by-Step Solution

Answer

The logarithm \( \log _{1 / 5} x \) expressed in common logarithms is \( -\frac{\log_{10} x}{\log_{10} 5 } \).

Step 1: Change of Base Formula

Let's begin our solution process by applying the change of base formula. Here, we want to express the given logarithm in terms of common logarithms. That is, we want to use base 10. Hence, we can rewrite the logarithm \( \log _{1 / 5} x \) as \( \frac{\log_{10} x}{\log_{10} ( 1 / 5 )} \).

Step 2: Simplify the Denominator

Now let's simplify the denominator of the fraction. We know that \( \log_{10} ( 1 / 5 ) \) is equal to \( - \log_{10} 5 \). Because, according to the properties of logarithms, \( \log_{p} (1 / q) = - \log_{p} q \). So, our fraction becomes \( \frac{\log_{10} x}{- \log_{10} 5 } \).

Step 3: Further Simplification

The last step is to simplify the fraction further. The denominator is negative, and when we divide by a negative number, the sign of the fraction changes. So the final expression is \( -\frac{\log_{10} x}{\log_{10} 5 } \).

Key Concepts

Common Logarithms

When we talk about common logarithms, we are referring to logarithms that have a base of 10. These are the logs that you often see without a base indicated, like this: \( \log x \). If you don't see a base, you can assume it's 10. This base 10 system is prevalent because of its connection to our decimal number system, making calculations more intuitive.

In the context of the exercise, the log base 10 is used to convert a logarithm of a different base into a more familiar form. The change of base formula allows us to take any \( \log_{b} x \) and rewrite it in terms of base 10, like \( \frac{\log_{10} x}{\log_{10} b} \), which can be especially useful in simplifying and comparing different logarithmic expressions.

Properties of Logarithms

The properties of logarithms are powerful tools that allow for the simplification and transformation of logarithmic expressions. There are several key properties that are particularly helpful:

• The Product Property: \( \log_{b}(xy) = \log_{b}x + \log_{b}y \). This property tells us that the log of a product is the sum of the logs.
• The Quotient Property: \( \log_{b}(\frac{x}{y}) = \log_{b}x - \log_{b}y \). This means the log of a quotient is the difference of the logs.
• The Power Property: \( \log_{b}(x^y) = y \cdot \log_{b}x \). This indicates that the log of an exponentiated term can be brought out as a multiplier.

These properties help drastically in breaking down complex logarithms into simpler parts that are easier to calculate or compare, as demonstrated in the exercise where the Quotient Property was applied to express \( \log_{10} ( 1 / 5 ) \) as \( -\log_{10} 5 \).

Logarithmic Simplification

The goal of logarithmic simplification is to make complex logarithmic expressions more manageable by utilizing various logarithmic properties and operations. In the given exercise, we applied logarithmic simplification by first using the change of base formula to convert to common logarithms and then by applying the properties of logarithms to simplify the expression.

For instance, when the denominator of a fractional logarithmic expression is negative, as was the case with \( \log_{10} ( 1 / 5 ) \), understanding that division by a negative flips the sign of the fraction is crucial in the simplification process. Simplifications like this help in two ways: they make it easier to understand the behavior of the logarithmic function and they often enable further calculations with the expression, such as solving logarithmic equations or comparing logarithmic values.