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Chapter 10: Problem 27

Evaluate each logarithm to four decimal places. $$ \ln 10 $$

Short Answer

Expert verified
2.3026

Step by step solution

01

Understand the Natural Logarithm

The natural logarithm function, denoted as \(\text{ln}\), is the logarithm to the base \(e\), where \(e\) is approximately equal to 2.71828. We need to find the value of \(\text{ln}(10)\).
02

Use a Calculator

To evaluate \(\text{ln}(10)\) to four decimal places, input the expression into a scientific calculator. Make sure the calculator is set to provide results to at least four decimal places.
03

Record the Result

After inputting \(\text{ln}(10)\) into the calculator, the value is approximately 2.3026.

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Key Concepts

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

logarithms
Logarithms are mathematical operations that help in solving equations involving exponents. A logarithm answers the question: 'To what exponent must a given base be raised, to produce a certain number?' For example, in the equation \(\text{log}_{a}(b) = c\), \(a^{c} = b\).

The natural logarithm, denoted by \(\ln\), is a special type of logarithm where the base is \(e\) (Euler's number, approximately 2.71828).

This makes it particularly useful in various scientific and engineering calculations.
base e
The letter \(e\) represents Euler's number, an irrational constant approximately equal to 2.71828.

It is the base of the natural logarithm. Its unique properties make it valuable in continuous growth or decay problems, such as population growth or radioactive decay.

When you see \(\ln\), it means you are working with logarithms with base \(e\). This base is central in calculus and complex exponential functions. The formula for the natural logarithm of a number is \(\ln(x)\), which is the exponent to which \(e\) must be raised to produce \(x\).
scientific calculator
Using a scientific calculator is essential when dealing with natural logarithms, especially for values not easily solved by hand.

To find \(\ln(10)\) to four decimal places:
  • Turn on the calculator
  • Navigate to the \(\ln\) function
  • Input \(10\)
  • Press 'Enter' or '=' to get the result
You should get approximately 2.3026. Most scientific calculators will provide results to multiple decimal places, allowing for precise calculations.

Always ensure your calculator is set to display results to at least four decimal places for accurate data.

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Most popular questions from this chapter

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} \sqrt[4]{9} $$

Solve each equation. Approximate solutions to three decimal places. $$ 9^{-x+2}=13 $$

Solve each equation. Give exact solutions. $$ \log _{3} x+\log _{3}(2 x+5)=1 $$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$\log _{10} 2=0.3010 \text { and } \log _{10} 9=0.9542$$ Use these values and the properties of logarithms to evaluate each expression. DO NOT USE A CALCULATOR. See Example 5. $$ \log _{10} 18 $$

Solve each equation. Approximate solutions to three decimal places. $$ 6^{x+3}=4^{x} $$
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