/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 52 Use a calculator to evaluate eac... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 52

Use a calculator to evaluate each expression. Round your answer to three decimal places. $$ \frac{\ln 5}{3} $$

Short Answer

Expert verified
0.536

Step by step solution

01

Understand the Function

The expression to evaluate is \(\frac{\ln 5}{3}\). This means you need to find the natural logarithm of 5 and then divide the result by 3.
02

Use a Calculator to Find the Natural Logarithm

On your calculator, locate the natural logarithm function, usually labeled as 'ln'. Enter 5 and press the 'ln' button to get \(\ln 5\). The value should be approximately 1.609.
03

Divide the Result by 3

Take the value of \(\ln 5 \), which is approximately 1.609, and divide it by 3. Use the division function on your calculator to get \(\frac{1.609}{3} \).
04

Round to Three Decimal Places

After performing the division, the result is approximately 0.536333. Round this value to three decimal places to get 0.536.

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Key Concepts

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

Natural Logarithm
The natural logarithm, denoted as \(\text{ln}\), is a special type of logarithm. It uses the mathematical constant e (approximately 2.718) as its base. This is different from common logarithms that use 10 as their base.
When you see \(\text{ln}(x)\), it is asking you for the power to which e must be raised to result in x. For example, \(\text{ln}(1)\ = 0\), because e raised to the power of 0 is 1. Similarly, \(\text{ln}(e)\ = 1\), because e raised to the power of 1 is e itself.
In the original exercise, we needed to find \(\text{ln}(5)\), which is the power to which e must be raised to get approximately 5. Using a calculator, this value was found to be approximately 1.609.
Logarithmic Functions
Logarithmic functions are the inverses of exponential functions. This means that if you have an exponentiation like \[a^b = c\text{, you can write the equivalent logarithmic form as }\ \text{log}_a(c) = b.\] The base here, a, can be any number, but e is a common and important base in naturallogarithms.
Key properties of logarithmic functions include:
  • \(\text{log}_a(1) = 0\)
  • \(\text{log}_a(a) = 1\)
  • \(\text{log}_a(x \times y) = \text{log}_a(x) + \text{log}_\text{a}(y)\)
  • \(\text{log}_a(\frac{x}{y}) = \text{log}_a(x) - \text{log}_a(y)\)
.
These properties help in simplifying and solving logarithmic equations and expressions. In our exercise, we used the natural logarithm, which is a specific case of these general logarithmic functions.
Numerical Methods
Numerical methods involve techniques to approximate mathematical procedures. They are especially helpful when a formula doesn鈥檛 directly solve the problem or when exact solutions are not possible.
Calculators use numerical methods to compute values like natural logarithms. Specific numerical methods include:
  • Newton's Method: Useful for finding successively better approximations to the roots (or zeroes) of a real-valued function.
  • Interpolation Methods: Used to estimate unknown values that fall between known values.
  • Series Expansion: Natural logarithms can be calculated using series expansions like the Taylor or Maclaurin series.
.
In our exercise, the calculator used internal numerical methods to quickly and efficiently find \(\text{ln}(5)\), which we divided by 3 to get the final result.

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Most popular questions from this chapter

Find the value of \(\log _{2} 3 \cdot \log _{3} 4 \cdot \log _{4} 5 \cdot \log _{5} 6 \cdot \log _{6} 7 \cdot \log _{7} 8\).

Two bank accounts are opened at the same time. The first has a principal of $$ 1000$ in an account earning 5 % compounded monthly. The second has a principal of 2000 in an account earning 4 % interest compounded annually. Determine the number of years, to the nearest tenth, at which the account balances will be equal.

Show that \(\log _{a}(\sqrt{x}+\sqrt{x-1})+\log _{a}(\sqrt{x}-\sqrt{x-1})=0\)

Suppose that \(G(x)=\log _{3}(2 x+1)-2\) (a) What is the domain of \(G ?\) (b) What is \(G(40) ?\) What point is on the graph of \(G ?\) (c) If \(G(x)=3,\) what is \(x ?\) What point is on the graph of \(G ?\) (d) What is the zero of \(G ?\)

Solve each equation. $$ e^{-2 x}=\frac{1}{3} $$
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