Problem 48 - Calculus and its applications

Question

Find each logarithm. Round to six decimal places. $$ \ln 8100 $$

Step-by-Step Solution

Answer

\( \ln 8100 \) is approximately 9.000353.

Step 1: Understanding the Natural Logarithm

The natural logarithm, denoted as \( \ln \), is the logarithm to the base \'e\', where \'e\' is an irrational constant approximately equal to 2.7182818. Thus, \( \ln 8100 \) asks us to find the power to which we raise \'e\' to obtain 8100.

Step 2: Use a Calculator

To calculate \( \ln 8100 \), you should use a scientific calculator. Enter 8100 and press the \( \ln \) function key to get the natural logarithm. Ensure that your calculator is set to the correct mode (usually scientific) to obtain an accurate result.

Step 3: Record the Result

After inputting the value and applying the \( \ln \) function on your calculator, you should get a result of approximately 9.000353.

Step 4: Round to Six Decimal Places

The exercise asks to round the result to six decimal places. Record the result as 9.000353, ensuring that it has six decimal places as required.

Key Concepts

Logarithms

Logarithms are a fundamental concept in mathematics that help us solve equations involving exponential growth or decay. At its core, a logarithm is the inverse of exponentiation. This means if you have a number being raised to an exponent to get another number, the logarithm allows you to find that exponent.

• For example, if you know that \( a^b = c \), then \( \log_a(c) = b \). Here, \( \log_a \) is the logarithm with base \( a \) and it tells you the power \( b \).

The natural logarithm, denoted as \( \ln \), specifically uses the number \( e \), an irrational number roughly 2.718, as its base. This is common in many fields such as science and engineering where continuous growth processes are studied. Logarithms are incredibly useful for simplifying complex calculations and solving for unknowns in exponential models.

Scientific Calculator

A scientific calculator is a vital tool for anyone dealing with mathematical problems involving functions like logarithms, trigonometry, or calculus. Unlike basic calculators, scientific calculators are equipped to handle more complex functions, including natural logarithms, square roots, and powers. Here’s how to use one effectively:

• Ensure that your calculator is in the correct mode. For logarithms, especially natural logarithms, this is generally the default scientific mode.
• To find a natural logarithm, input the number in question and press the \( \ln \) button. For instance, for \( \ln 8100 \), you will type 8100 followed by the \( \ln \) key.
• Make sure your calculator’s display is set to show enough decimal places if precision is required. For our example, round your answer to six decimal places as asked by the problem.

Using scientific calculators not only helps you get precise answers but also saves time, especially when handling large numbers or complex operations.

Irrational Numbers

Irrational numbers are numbers that cannot be expressed as a simple fraction of two integers. Unlike rational numbers, their decimal representation doesn't terminate or repeat. A classic example of an irrational number is \( \pi \), but another crucial one in mathematics is \( e \), the base of the natural logarithm.

• \( e \) is approximately 2.7182818, and it is non-repeating, non-terminating.

These numbers are critical because they show up naturally in many real-world applications, like calculating compound interest, understanding exponential growth and decay, and in various mathematical models. Understanding irrational numbers helps you grasp why certain calculations, like those involving natural logs, may seem complex but are grounded in fundamental mathematical concepts.