Problem 2 - College Algebra and Calculus: An Applied Approach

Question

Write the logarithm in terms of common logarithms.\(\log _{7} 12\)

Step-by-Step Solution

Answer

Applying the change of base formula, \(\log _{7} 12\) can be written as \(\frac{\log_{10} 12}{\log_{10} 7}\).

Step 1: Identify logarithm components

In the logarithmic expression \(\log _{7} 12\), the base is 7 and the argument is 12.

Step 2: Apply change of base formula

By using the change of base formula, the base of the log can be changed to 10. With 'a' as 7, 'b' as 12, and 'c' as 10, this yields: \(\log_{7} 12 = \frac{\log_{10} 12}{\log_{10} 7}\).

Step 3: Calculation

The value of both \(\log_{10} 12\) and \(\log_{10} 7\) can be calculated using a calculator. This provides a numerical approximation of the original expression.

Key Concepts

Change of Base Formula

The change of base formula is a powerful mathematical tool. It allows us to write logs in a different base. Specifically, it converts a logarithm in any base to common logarithms (base 10). For the log
\( \log_{\text{base}}\), with base 7 as in our exercise, we can express it through the formula:

• For \(\log_{7} 12 = \frac{\log_{10} 12}{\log_{10} 7}\)

This means we divide the log of the argument (12) by the log of the base (7), using base 10 for both.
This transformation simplifies working with logarithms in different bases. It's useful when you have a calculator that only handles common logarithms.

Common Logarithms

Common logarithms are logs with base 10. They are written as \(\log_{10} \) or simply \(\log\). This base is popular since it aligns with the decimal system, which is the most widely used.

• For example, \(\log(100) = 2\) because 10 squared equals 100.

In the exercise, when we convert \(\log_{7} 12\) using the change of base formula, we express it in terms of common logs:

• The expression becomes \(\log_{10} 12\) and \(\log_{10} 7\).

Using a calculator, you can easily evaluate these to get a numerical representation.

Numerical Approximation

Numerical approximation helps us find a close estimate of the result, particularly when exact solutions aren't feasible. Once the log expression is simplified into common logs, its values can be computed with a calculator.

• For \(\log_{10} 12\), the calculator gives a decimal value.
• Similarly, \(\log_{10} 7\) yields another decimal.

Once you have these values, divide one by the other as per the change of base formula:

• This division gives the approximation of \(\log_{7} 12\).

Approximations are useful in many real-world applications, making complex calculations manageable.