Problem 50 - Precalculus

Question

Approximate (if possible) the common and natural logarithms using a calculator. Round to two decimal places. $$\log 380$$

Step-by-Step Solution

Answer

The common logarithm of 380 is approximately 2.58.

Step 1: Understanding the Logarithm

The logarithm function, \( \log \), is used to find the power to which a base number must be raised to obtain another number. In this exercise, we need to calculate the common logarithm of 380. This means we need to find \( \log 10^x = 380 \). The common logarithm function uses a base of 10.

Step 2: Use a Calculator

Since 380 is not a simple power of 10, we will use a calculator to find the logarithm. Enter the number 380 into the calculator and apply the common logarithm function (log base 10) to find its value.

Step 3: Recording the Logarithm

After inputting \( \log 380 \) into the calculator, we obtain the logarithm value. The calculator will provide the result to multiple decimal places. Round this result to two decimal places.

Step 4: Rounding the Logarithm

The calculator gives the value of \( \log 380 \approx 2.579783 \). Rounding this to two decimal places gives us \( 2.58 \).

Key Concepts

Understanding the Common Logarithm

The concept of a common logarithm is fundamental in mathematics and plays a vital role in simplifying complex calculations. A common logarithm is a logarithm that uses base 10. This means any number can be expressed as a power of 10, which is especially useful in dealing with large or small numbers, much like scientific notation. Imagine you want to know the power to which 10 should be raised to get 380, that's where the common logarithm kicks in. The notation for a common logarithm is simply \( \log \), implying base 10 by default. Therefore, when you see \( \log 380 \), it's asking for the exponent of 10 that equals 380. Calculators have a built-in function to compute this quickly.
Here are a few additional points:

• The equation \( 10^x = 380 \) requires finding \( x \), which equals \( \log 380 \).
• Common logarithms are widely used in various fields, such as engineering, science, and finance, due to their simplicity when dealing with exponential growth or decay.
• Always round your results to the required decimal places, ensuring precision and uniformity in your calculations.

Exploring the Natural Logarithm

Natural logarithms are another form of logarithms, distinct in their base rather than their function. The base for natural logarithms is the mathematical constant \( e \) (approximately 2.718), and they are depicted as \( \ln \). This is the logarithm base \( e \), widely used in calculus and mathematical modeling of growth processes.
Why do we use \( e \)? It is uniquely interesting because \( e \) represents continuous growth and appears naturally in various mathematical contexts like compound interest calculations, population dynamics, and differential equations.
Some important points to consider:

• Natural logarithms answer the question: "To what power must \( e \) be raised, to equal a given number?"
• Just like with common logarithms, you can use a calculator to find \( \ln x \).
• Natural logarithms connect deeply with growth rates and can simplify the calculations in exponential functions.

While the original exercise focused on the common logarithm, understanding both can greatly enhance your mathematical literacy.

Key Tips for Calculator Rounding

Rounding is crucial when dealing with decimal results from calculators, ensuring the final figures are practical and understandable. In mathematics, precision is key, and often, decisions must be made on how many decimal places to keep.
Here's why rounding matters:

• It provides a consistent way to present decimal data, which is essential in scientific and technical documentation.
• Avoids overcomplicating results with unnecessary digits that may not impact the overall understanding of a computation.
• Rounding to two decimal places is common practice when dealing with logarithms because it offers a balance between precision and simplicity.

To round effectively:

• Identify which decimal place you need to round to—commonly the second place for logarithms.
• Look at the digit that follows the last intended decimal place. If it's 5 or higher, round up. If it's less than 5, keep the digit as is.

For instance, if your calculator displays \( \log 380 \approx 2.579783 \), rounding to two decimal places provides \( 2.58 \). By following such straightforward rules, your results will be both accurate and easy to interpret.