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Chapter 12: Problem 17

Find each logarithm. Give approximations to four decimal places. \(\ln 0.0556\)

Short Answer

Expert verified
\(\ln 0.0556 \approx -2.8904\)

Step by step solution

01

- Understanding the Problem

The task is to find the natural logarithm (\(\ln\)) of 0.0556. A natural logarithm uses the base \(e\), where \(e\approx2.71828\).
02

- Use a Calculator

To find the value of \(\ln(0.0556)\), use a scientific calculator. Enter 0.0556 and then press the \(\ln\) function button.
03

- Calculate and Round

The calculator will give you a value for \(\ln(0.0556)\). Round this value to four decimal places. The approximate value is \(\approx -2.8904\).

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Key Concepts

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

logarithms
Logarithms are an essential concept in mathematics. A logarithm answers the question: 鈥楾o what exponent must we raise a certain base, to obtain a given number?鈥�. For example, in the expression \(\text{ln}(x)\), 'ln' stands for natural logarithm, which means we use base \(e\). Logarithms help in simplifying multiplication and division into addition and subtraction. They are widely used in many scientific, engineering, and statistical calculations. Understanding logarithms requires grasping their base and the transformation they imply.
base e
The constant base \(e\) (approximately 2.71828) is a crucial number in mathematics, especially in logarithms. It is the base for the natural logarithm. The number \(e\) has unique properties, making it indispensable in calculus, complex analysis, and solving differential equations. It often appears in situations involving exponential growth or decay, such as population increase or radioactive decay.
scientific calculator
A scientific calculator is necessary for computing logarithms accurately. To find \(\text{ln}(0.0556)\), follow these steps:
  • Turn on your calculator and make sure it is in the correct mode (degree/radian if required).
  • Enter the value 0.0556.
  • Press the 'ln' key to get the natural logarithm.
. A scientific calculator helps streamline the process, providing quick, reliable results for complex operations such as logarithms.
rounding to four decimal places
Rounding to four decimal places means adjusting your result such that only four digits are kept after the decimal point. This is crucial for precision in scientific calculations. For example, when you find that \(\text{ln}(0.0556)\) = -2.89038584, rounding it to four decimal places gives you -2.8904. Rounding rules state if the fifth decimal place is 5 or higher, you round up the fourth decimal place by 1. Otherwise, you leave it as is.

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Most popular questions from this chapter

Consumers can now enjoy movies at home in elaborate home-theater systems. Find the average decibel level $$D=10 \log \left(\frac{I}{I_{0}}\right)$$ for each movie with the given intensity \(I\) (a) Avatar; \(5.012 \times 10^{10} I_{0}\) (b) Iron Man \(2 ; \quad 10^{10} I_{0}\) (c) Clash of the Titans; \(6,310,000,000 \mathrm{I}_{0}\)

Simplify each expression. Write answers using only positive exponents. See Sections 4.1 and 4.2. $$ \frac{7^{8}}{7^{-4}} $$

The asymptote of the graph of \(f(x)=a^{x}\) A. is the \(x\) -axis B. is the \(y\) -axis C. has equation \(x=1\) D. has equation \(y=1\)

Each function defined is one-to-one. Find the inverse algebraically, and then graph both the function and its inverse on the same graphing calculator screen. Use a square viewing window. \(f(x)=\sqrt[3]{x+2}\)

If \(f(x)=4^{x},\) find each value indicated. In Exercise 50, use a calculator, and give the answer to the nearest hundredth. \(f\left(-\frac{1}{2}\right)\)
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