Question

Use \(\log _{a} x=(\ln x) /(\ln a)\) to calculate each of the logarithms. $$ \log _{7}(0.11) $$

Short answer

\( \log_{7}(0.11) \approx -1.134 \)

Step-by-step solution

Identify the formula components

The given formula is \( \log_{a} x = \frac{\ln x}{\ln a} \). In the exercise, we identify \( a = 7 \) and \( x = 0.11 \).

Calculate \( \ln(0.11) \)

Use a calculator to find the natural logarithm of 0.11. The result is approximately \( \ln(0.11) \approx -2.2073 \).

Calculate \( \ln(7) \)

Use a calculator to find the natural logarithm of 7. The result is approximately \( \ln(7) \approx 1.9459 \).

Apply the formula for \( \log_{7}(0.11) \)

Using the formula \( \log_{7}(0.11) = \frac{\ln(0.11)}{\ln(7)} \), substitute the values obtained: \( \frac{-2.2073}{1.9459} \approx -1.134 \).

Key concepts

Natural Logarithm

The natural logarithm, often denoted as \( \ln \), is a special type of logarithm because it uses the base \( e \), where \( e \) is approximately equal to 2.71828. It is widely used in mathematics, especially in calculus and complex calculations.
Understanding the concept of the natural logarithm simplifies many mathematical processes:

• Natural logs transform exponential functions, making them more manageable.
• They have nice properties that make them useful for solving exponential equations.
• The derivative of \( \ln(x) \) is simple, which is just \( 1/x \), facilitating usage in calculus.

Thus, natural logarithms are not just theoretical constructs; they have practical applications in various scientific fields, such as physics, chemistry, and statistics, helping solve complex exponential growth problems.

Change of Base Formula

The change of base formula is a powerful tool when you need to compute logarithms with a base that you don't have a calculator function for. This formula is expressed as:\[ \log_{a}(x) = \frac{\ln(x)}{\ln(a)}\]Here, \( \ln \) represents the natural logarithm. This equation allows you to convert any logarithm into terms of natural logarithms, which can then be easily calculated using common calculators.
The main idea behind the change of base formula is to express any logarithm in terms of another base. Useful points include:

• Converts Logarithms: If your calculator only supports natural logs, you can still calculate base-\( a \) logs.
• Universal Application: It works for any positive real numbers \( a \) and \( x \), as long as \( a eq 1 \).
• Simplifies Calculations: Particularly useful in solving real-world problems where specific log bases are needed.

Employing the change of base formula can unlock a broader range of logarithmic functions and ensure consistency across different mathematical calculations.

Logarithmic Calculations

Performing logarithmic calculations requires understanding the mechanisms of transforming and manipulating these functions. In the provided exercise, the focus is calculating \( \log_{7}(0.11) \) using the change of base formula.
The process involved several key steps:- Identify Components: Recognize \( a \) as the base and \( x \) as the value.- Compute Natural Logs: Use a calculator to find \( \ln(x) \) and \( \ln(a) \).- Apply Formula: Substitute these values into \( \log_{a}(x) = \frac{\ln(x)}{\ln(a)} \).Understanding these steps makes logarithmic calculations more systematic:

• Precision: Ensure that computations are accurate by checking over each step, particularly the use of the calculator for \( \ln \) values.
• Verification: Cross-verify results to confirm the accuracy of the calculated logarithm.
• Application in Real-Life: Logarithms are used in measuring intensity (decibels in sound), Richter scale for earthquakes, and other scientific datasets.

By mastering these calculations, students can solve more complex problems efficiently and inspire confidence in handling various mathematical scenarios.