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Chapter 5: Problem 51

Use a calculator to find a decimal approximation for each common or natural logarithm. $$\ln 43$$

Short Answer

Expert verified
\( \ln 43 \approx 3.7612 \)

Step by step solution

01

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 2.71828. We want to find the value of \( \ln 43 \), which is asking, "To what power must we raise \( e \) to get 43?"
02

Using the Calculator

Turn on your calculator and ensure that it is in scientific mode, as this allows for the calculation of logarithms. Find the button marked \( \ln \) on your calculator. Type "43" and then press the \( \ln \) button. This will give you the natural logarithm of 43.
03

Reading the Result

After pressing the \( \ln \) button, your calculator should display the result. The decimal approximation of \( \ln 43 \) is approximately 3.7612. It is common to round this number to four decimal places unless otherwise specified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Decimal Approximation
When working with logarithms, especially the natural logarithm like \( \ln 43 \), it's important to express the result as a decimal approximation. This means representing an exact number as closely as possible using decimals. Since \( e \) (Euler's number) is irrational, exact calculations can be cumbersome. Thus, we use decimal approximations.
  • Decimal approximations allow us to practically express irrational numbers or complex computations in a simpler form.
  • In most calculations, you'll aim to approximate to 4 decimal places. For instance, \( \ln 43 \approx 3.7612 \).
  • These approximations are useful in real-world applications where exact values are either unnecessary or difficult to use.
Utilizing decimal approximations helps in making the numbers more manageable and understandable, thereby simplifying calculations in various fields like engineering and physics.
Base e
The concept of the natural logarithm revolves around \( e \), an irrational and transcendental number which is approximately 2.71828. Being the foundation of natural logarithms, \( e \) plays a crucial role in calculus and complex analysis.
  • Euler's number, \( e \), serves as the base for natural logarithms, making it a cornerstone in mathematical computations.
  • Logarithms with base \( e \) are termed as "natural" owing to their occurrence in natural exponential growth processes.
  • The equation \( \ln 43 \) asks for the power \( e \) needs to be raised to achieve 43, revealing the inherent connection between exponential and logarithmic functions.
This base \( e \) underpins many phenomena in nature and finance, including compound interest and population growth.
Scientific Calculator
A scientific calculator is an invaluable tool for calculating expressions like \( \ln 43 \), especially when working with exponential functions and logarithms. Understanding your calculator's functionalities can significantly ease such computations.
  • Turn on your calculator and switch to scientific mode to access advanced functions like logarithms.
  • Look for the \( \ln \) button; it's specifically designed to compute natural logarithms.
  • Enter the number for which you wish to find the logarithm (like 43), press \( \ln \), and the calculator will provide the result, often rounded to several decimal places.
Using a scientific calculator streamlines these processes, aiding in quick computations essential in more intricate scientific or mathematical analysis.

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Most popular questions from this chapter

Newton's law of cooling says that the rate at which an object cools is proportional to the difference \(C\) in temperature between the object and the environment around it. The temperature \(f(t)\) of the object at time t in appropriate units after being introduced into an environment with a constant temperature \(T_{0}\) is $$f(t)=T_{0}+C e^{-k t}$$ where \(C\) and \(k\) are constants. Use this result. A piece of metal is heated to \(300^{\circ} \mathrm{C}\) and then placed in a cooling liquid at \(50^{\circ} \mathrm{C}\). After 4 minutes, the metal has cooled to \(175^{\circ} \mathrm{C}\). Estimate its temperature after 12 minutes.

Suppose \(f(x)\) is the number of cars that can be built for \(x\) dollars. What does \(f^{-1}(1000)\) represent?

Restrict the domain so that the function is one-to-one and the range is not changed. You may wish to use a graph to help decide. Answers may vary. $$f(x)=-\sqrt{x^{2}-16}$$

Growth of E. coli Bacteria \(\mathrm{A}\) type of bacteria that inhabits the intestines of animals is named \(E .\) coli (Escherichia coli ). These bacteria are capable of rapid growth and can be dangerous to humans- especially children. In one study, \(E .\) coli bacteria were capable of doubling in number every 49.5 minutes. Their number after \(x\) minutes can be modeled by the function $$ N(x)=N_{0} e^{0.014 x} $$ (Source: Stent, G. S., Molecular Biology of Bacterial Viruses, W. H. Freeman.) Suppose \(N_{0}=500,000\) is the initial number of bacteria per milliliter. (a) Make a conjecture about the number of bacteria per milliliter after 99 minutes. Verify your conjecture. (b) Estimate graphically the time elapsed until there were 25 million bacteria per milliliter.

Use any method (analytic or graphical) to solve each equation. $$\log _{2} \sqrt{2 x^{2}}-1=0.5$$
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