Problem 56
Question
Use a calculator to evaluate each expression, if possible. Express all answers to four decimal places. See Using Your Calculator: Evaluating Base-e (Natural) Logarithms. $$\ln 378.96$$
Step-by-Step Solution
Verified Answer
The natural logarithm of 378.96 is approximately 5.9372.
1Step 1: Understanding Natural Logarithm
The natural logarithm, denoted as \( \ln \), is the logarithm to the base \( e \), where \( e \) is approximately equal to 2.71828. Our goal is to find the natural logarithm of the given number, 378.96.
2Step 2: Input into Calculator
Turn on your calculator and ensure it is in the standard mode (not scientific or programming mode). Locate the 'ln' button on your calculator. This button is often labeled as 'ln' and represents the natural logarithm function.
3Step 3: Enter the Number
Press the 'ln' button, then enter the number 378.96. Your calculator should now display something like \( \ln(378.96) \).
4Step 4: Calculate the Result
Press the 'equals' button (often 'Enter' or '=') to calculate the result. The calculator will compute the natural logarithm of 378.96.
5Step 5: Rounding to Four Decimal Places
Check the result displayed on the calculator. It should be a decimal. Round this result to four decimal places. If the fifth decimal place is 5 or more, round the fourth decimal place up by one.
Key Concepts
base eusing a calculatordecimal places
base e
The natural logarithm, represented by \( \ln \), is a special type of logarithm where the base is \( e \). But what exactly is \( e \)? The base \( e \) is a mathematical constant approximately equal to 2.71828. It’s a fascinating number used extensively in mathematics, especially in calculus and complex analysis.
\( e \) is called the "Euler's number," named after the Swiss mathematician Leonhard Euler. It's like how we often use 10 as the base for our common logarithms, known as base-10 or "log". But here, \( e \) serves a similar role for natural logarithms. Why is it special? Well, \( e \) appears frequently in problems involving growth and decay, like calculating compound interest or modeling population growth.
Remembering that \( e \) is approximately 2.71828 can help you understand why the natural logarithm is sometimes more suitable in scientific calculations. So, when you see \( \ln(378.96) \), it simply means you are finding the logarithm of 378.96 with the base \( e \).
\( e \) is called the "Euler's number," named after the Swiss mathematician Leonhard Euler. It's like how we often use 10 as the base for our common logarithms, known as base-10 or "log". But here, \( e \) serves a similar role for natural logarithms. Why is it special? Well, \( e \) appears frequently in problems involving growth and decay, like calculating compound interest or modeling population growth.
Remembering that \( e \) is approximately 2.71828 can help you understand why the natural logarithm is sometimes more suitable in scientific calculations. So, when you see \( \ln(378.96) \), it simply means you are finding the logarithm of 378.96 with the base \( e \).
using a calculator
Calculators have made solving mathematical problems much easier, and evaluating natural logarithms using one is a breeze once you know the steps. To start, ensure your calculator is in the standard calculation mode, not set to something like scientific or programming. Then, look for the "ln" button – this is crucial, as it's specifically designed to compute the natural logarithm.
Here's how you can do it:
Here's how you can do it:
- Switch on your calculator.
- Locate and press the "ln" button – this opens the function needed to compute the natural logarithm.
- Next, type in the number you want to find the natural logarithm for, like 378.96.
- Your calculator will show something like \( \ln(378.96) \).
decimal places
Understanding decimal places is crucial for ensuring precision in calculations. When a problem asks you to express an answer to four decimal places, you are to report the number showing exactly four digits to the right of the decimal point.
For instance, if your calculator displays a result like 5.937284 after calculating \( \ln(378.96) \), you'll need to round it to four decimal places. Here’s how you do it:
For instance, if your calculator displays a result like 5.937284 after calculating \( \ln(378.96) \), you'll need to round it to four decimal places. Here’s how you do it:
- Identify the fourth decimal position. In this example, it's the number 7.
- Look at the fifth decimal position, which is 2.
- If this number, the fifth one, is 5 or higher, you'll round the fourth decimal place up by one. If it’s less, you leave the fourth decimal unchanged.
Other exercises in this chapter
Problem 55
Show that each pair of functions are inverses. \(f(x)=2 x+9, f^{-1}(x)=\frac{x-9}{2}\)
View solution Problem 55
Write logarithm without an exponent or a radical symbol. Then simplify, if possible. \(\log e^{-3}\)
View solution Problem 56
Solve each equation. See Example \(8 .\) $$\log _{2}(x-7)+\log _{2} x=3$$
View solution Problem 56
Find \(f(x)\) and \(g(x)\) such that \(h(x)=(f \circ g)(x) .\) Answers may vary. $$ h(x)=\frac{1}{3 x-16} $$
View solution