Question

Find the natural logarithm of each number. $$4.77$$

Short answer

The natural logarithm of 4.77 is approximately 1.5645.

Step-by-step solution

Identify the Problem

We are asked to find the natural logarithm of the number 4.77. The natural logarithm is denoted by ln, which corresponds to the logarithm with base e, where e is approximately 2.71828.

Use a Calculator

The natural logarithm of a number is not usually one that can be calculated by hand for non-integer numbers. Here, we will use a scientific calculator or logarithm table to find the value of ln(4.77). Input 4.77 into the calculator and use the natural logarithm function to get the result.

Record the Result

After using the calculator, we have found that the natural logarithm of 4.77 is approximately 1.5645 to four decimal places.

Key concepts

Logarithms

Logarithms are an essential concept in mathematics, particularly in algebra and calculus. Imagine you have an equation like \( b^x = y \), where you know the values of \( b \) and \( y \), but not \( x \). The logarithm helps us find \( x \), which is the power to which the base \( b \) must be raised to produce \( y \). In simpler terms, it answers the question: 'To what exponent must the base be raised to get a certain number?’

The natural logarithm, denoted as \( \ln \), is a special type of logarithm where the base is the mathematical constant \( e \). The equation \( \ln(y) = x \) means that \( e^x = y \).

Understanding natural logarithms is vital because they are used to model many natural phenomena such as growth and decay processes in biology, chemistry, and physics. For instance, the decay of radioactive substances and the growth of bacterial populations are often modeled using natural logarithms. Learning to work with logarithms, including using scientific calculators to solve them, is a valuable skill in many scientific and engineering fields.

Scientific Calculator

A scientific calculator is a handy tool for students and professionals who frequently deal with complex mathematical operations. Unlike standard calculators, scientific calculators can perform a variety of functions such as trigonometric calculations, exponentials, logarithms, and even statistical analysis.

To calculate a natural logarithm using a scientific calculator:

• Turn on the calculator.
• Enter the number you want to find the natural logarithm of. For the given exercise, that's 4.77.
• Look for a button labeled \( \ln \), which stands for natural logarithm.
• Press the \( \ln \) button after typing in your number. The display should show the natural logarithm of the entered number.

Calculators might differ slightly in operation, so if you are uncertain, consult the calculator's manual. Exercises like the one provided help establish familiarity with these functions, which is vital for efficiency in solving more complex problems.

Base e

The number \( e \) is an irrational mathematical constant approximately equal to 2.71828 and is one of the most important numbers in mathematics, similar to \( \pi \). It's known as Euler's number after the Swiss mathematician Leonhard Euler.

\( e \) is the base of the natural logarithm, which is why the natural logarithm of \( e \) itself is equal to 1 (i.e., \( \ln(e) = 1 \)). The significance of \( e \) arises from its unique properties in calculus, especially concerning the rate of growth. If a quantity grows at a rate proportional to its current size—a common scenario in interest calculations, population growth, and radioactive decay—then the function that describes this growth is an exponential function with base \( e \).

The simplicity of its derivative and integral properties makes \( e \) fundamental to mathematics, and thus, understanding \( e \) is crucial for studying any mathematical discipline involving calculus or complex growth processes.