Problem 9 - Algebra and Trigonometry

Question

Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$ \log \left(\frac{x}{100}\right) $$

Step-by-Step Solution

Answer

The expanded form of the given logarithmic expression is \(\log x - 2\).

Step 1: Use the Quotient Rule of Logarithms

The Quotient rule states that \(\log_b(a/c) = \log_ba - \log_bc\). So, \(\log(x/100)\) can be expanded to \(\log x - \log 100\)

Step 2: Evaluate Logarithm

Now evaluate \(\log 100\). Since the base of the logarithm is not specified, it is assumed to be 10 (common logarithm). The \(\log 100\) is 2 because 10 raised to 2 equals 100.

Step 3: Final Expression

Substitute the evaluated value from step 2 into the expression obtained in step 1 to get the final expanded expression. The final expanded expression therefore is \(\log x - 2\)

Key Concepts

Quotient Rule of Logarithms

The Quotient Rule of Logarithms is a useful property that helps simplify expressions where division inside a logarithm is involved. Essentially, it states that a logarithm of a quotient can be rewritten as a difference of logarithms. Mathematically, this rule is expressed as:

• \( \log_b\left(\frac{a}{c}\right) = \log_b a - \log_b c \)

So, if you're tackling a problem where you need to expand a logarithm involving division, apply this rule to break it down into two separate logarithms.

It's important because it transforms a potentially complex expression into a more manageable one, helping with computations and understanding the function's behavior. In our exercise, applying the quotient rule took the expression \( \log\left(\frac{x}{100}\right) \) and expanded it to \( \log x - \log 100 \).

Common Logarithm

A common logarithm is a logarithm with base 10. When no base is specified in a logarithmic expression, it is generally understood to be base 10. This is why common logarithms are often seen in scientific and technical calculations, where base 10 makes computations more straightforward. The notation \( \log(x) \) is typically used for common logarithms.

For example, when you encounter \( \log 100 \), and no base is mentioned, you can assume it means \( \log_{10} 100 \). In our specific problem, once we applied the quotient rule, we had to solve \( \log_{10} 100 \), which equals 2, since 10 squared is 100. Recognizing and evaluating common logarithms quickly aids in simplifying expressions further.

Expanding Logarithmic Expressions

Expanding logarithmic expressions is a technique used to simplify a complex logarithmic expression into simpler parts. This is essential, as it allows one to understand, analyze, and solve mathematical problems easily.

When expanding a logarithmic expression, several rules are applied, including the quotient, product, and power rules. The focus here is on the quotient rule, used to break down the division inside the logarithm into two separate parts.

• For example, \( \log\left(\frac{x}{100}\right) \) was expanded into \( \log x - \log 100 \).
• After applying the quotient rule, we simplify more by recognizing and calculating any known values, such as \( \log 100 \), which turns out to be 2.

By expanding, each part becomes more manageable to deal with or evaluate, leading to a simpler final expression, crucial for both theoretical understanding and practical applications.