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Chapter 12: Problem 16

Find each logarithm. Give approximations to four decimal places. \(\ln 8.32\)

Short Answer

Expert verified
ln(8.32) 鈮� 2.1203

Step by step solution

01

Understanding Natural Logarithms

Natural logarithms, denoted by \(\text{ln}\), are logarithms with the base \(e\), where \(e\) is approximately equal to 2.71828. This means \( \text{ln}(x) = y \) if and only if \( e^y = x \). We'll use a calculator to find the natural logarithm of 8.32.
02

Use a Calculator

Enter 8.32 into a scientific calculator. Press the \( \text{ln} \) button to compute its natural logarithm.
03

Read the Result

The calculator should display the result. For \( \text{ln}(8.32) \), the value is approximately 2.1203. Make sure to round the result to four decimal places if necessary.

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Key Concepts

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

logarithms
Logarithms are mathematical expressions that help us determine the power to which a certain base number must be raised to produce a given number. When we say \(\text{log}_b(a)\), we mean the power to which the base \(b\) must be raised to get \(a\). This is fundamental in many fields, including science and engineering. Logarithms can have any positive real number base, but the most common bases are 10 (common logarithms) and \(e\) (natural logarithms). The logarithm of a number is the inverse of exponentiation.
base e
The base \(_e\) is a special mathematical constant approximately equal to 2.71828. Named after the Swiss mathematician Leonhard Euler, \(_e\) is fundamental in the study of calculus and complex analysis. Natural logarithms use \(e\) as their base, and they are denoted by \(\ln\). For example, \(\ln (8.32)\) means finding the exponent \(y\) satisfying \(e^y = 8.32\). The constant \(e\) appears in various growth and decay processes, such as population growth, radioactive decay, and the compounding of interest.
scientific calculator
A scientific calculator is a robust tool designed to perform complex mathematical functions, including logarithms, exponents, and trigonometric functions. To find a natural logarithm using a scientific calculator, follow these steps:
  • Turn on your calculator.
  • Enter the number whose natural logarithm you want to find鈥攊n this case, 8.32.
  • Press the \(\ln\) button to compute the natural logarithm.
The calculator will display the result, which can be approximated to the desired number of decimal places.
rounding numbers
Rounding numbers is a method used to simplify numbers by reducing the digits but keeping the value close to the original. To round a number to four decimal places, locate the fourth digit after the decimal point.
  • If the fifth digit is 5 or higher, increase the fourth digit by one.
  • If the fifth digit is less than 5, keep the fourth digit as is.
For example, when we compute \(\ln(8.32)\), the calculator might show 2.120294. To round this to four decimal places, we look at the fifth digit, which is 4 (less than 5), so the rounded result will be 2.1203. This makes it easier to report and use the number in further calculations.

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Most popular questions from this chapter

Which equation defines a one-to-one function? Explain why the others are not, using specific examples. A. \(f(x)=x\) B. \(f(x)=x^{2}\) C. \(f(x)=|x|\) D. \(f(x)=-x^{2}+2 x-1\)

The number of years, \(N(r),\) since two independently evolving languages split off from a common ancestral language is approximated by $$N(r)=-5000 \ln r$$ where \(r\) is the percent of words (in decimal form) from the ancestral language common to both languages now. Find the number of years (to the nearest hundred years) since the split for each percent of common words. (a) \(85 \% \text { (or } 0.85)\) (b) \(35 \% \text { (or } 0.35)\) (c) \(10 \% \text { (or } 0.10)\)

In your own words, describe the characteristics of the graph of an exponential function. Use the exponential function defined by \(f(x)=3^{x}\) (Exercise 5 ) and the words asymptote, domain, and range in your explanation.

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

What is the base in the expression \(\ln x ?\) A. \(e\) B. 1 C. 10 D. \(x\)
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