Problem 49 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use the properties of logarithms to expand the expression as a sum, difference, and/or constant multiple of logarithms. (Assume all variables are positive.)$$\log _{10} \frac{t}{8}$$.

Step-by-Step Solution

Answer

\(\log_{10} t - \log_{10} 8\)

Step 1: Identify log properties to use

First, note that the given logarithm is a log of a quotient. That means you can use the rule \(\log_b (M/N) = \log_b M - \log_b N\), and write \(\log_{10} (t/8)\) as \(\log_{10} t - \log_{10} 8 \).

Step 2: Evaluate Logarithm with constant

Look at the second term, \(\log_{10} 8\). Here, we must remember that \(\log_{10}\) is a common logarithm and it means that the base number is 10. Therefore, we determine the power we need to raise 10 to get 8, which unfortunately is not a nice integer. We thus leave it as \(\log_{10} 8\).

Step 3: Finalize the result

Keep the first log term as it is and since \(\log_{10} 8\) cannot be simplified further put it as it is: \(\log_{10} t - \log_{10} 8\).

Key Concepts

Logarithmic Expansion

The idea of logarithmic expansion is to break down a complex logarithmic expression into simpler parts. This technique uses the properties of logarithms to express a single logarithmic term as a combination of simpler terms. This can include sums, differences, or even constant multiples.

For example, if we have a logarithmic term like \( \log_b (M \cdot N) \), we can break it down using the product rule: \( \log_b M + \log_b N \). This makes it easier to manage and understand each component.

In the given exercise \( \log_{10} \frac{t}{8} \), we use the quotient rule (discussed more below) to expand it into \( \log_{10} t - \log_{10} 8 \). This expansion separates the log of the numerator and the log of the denominator, making each term more manageable.

Common Logarithms

Common logarithms are logarithms with base 10. They are often denoted as \( \log \) without explicitly showing the base. Since we often work in base 10 with our number system, common logarithms are widely used and understood.

In the exercise, \( \log_{10} t \) and \( \log_{10} 8 \) are both common logarithms. This means they are using base 10. To better grasp common logarithms, remember this simple rule: \( \log_{10} 100 = 2 \), because \( 10^2 = 100 \). However, some numbers like \( 8 \) do not equate easily to a nice integer power of 10. Thus, \( \log_{10} 8 \) remains in its approximate form unless solved using a calculator.

Quotient Rule for Logarithms

The quotient rule for logarithms is a fundamental property used to expand logarithmic expressions involving division. The rule states: \( \log_b \left( \frac{M}{N} \right) = \log_b M - \log_b N \). This means that when you take the log of a quotient, you can separate it into the log of the numerator minus the log of the denominator.

In the problem \( \log_{10} \frac{t}{8} \), we directly apply this rule. We identify the expression as a log of a quotient, allowing us to expand it to \( \log_{10} t - \log_{10} 8 \). This property is particularly useful because it simplifies working with logs, making them easier to handle and analyze in mathematical problems.