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Chapter 12: Problem 27

Evaluate each logarithm to four decimal places. \(\ln 10\)

Short Answer

Expert verified
2.3026

Step by step solution

01

Identify the logarithm type

This problem requires the evaluation of a natural logarithm, denoted as \(\text{ln} 10\). Natural logarithms use the base \(e\) (approximately 2.71828).
02

Use a calculator

To find the value of \(\text{ln} 10\), a scientific calculator can be used to evaluate the expression with high precision. Input \(\text{ln} 10\) into the calculator.
03

Round to four decimal places

After obtaining the result from the calculator, round the answer to four decimal places. The precise value obtained is approximately 2.302585, so rounding it to four decimal places results in 2.3026.

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Key Concepts

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

Understanding Logarithms
Logarithms are the inverses of exponential functions. If you have a number expressed in the form of an exponent, taking the logarithm helps you find the exponent itself. For example, if we know that 10 is equal to some base raised to an exponent, the logarithm tells us what that exponent is. In simple terms, if you have \(\text{log}_b a\), this means 'to what power must b be raised, to get a?'.
Here are some key points to remember about logarithms:
  • Logarithm base 10 is called the common logarithm and is denoted as \(\text{log}_{10}\).
  • Logarithm base e is called the natural logarithm and is denoted as \(\text{ln}\).
  • The inverse property: \(\text{log}_b(b^x) = x\) for any base, b.
Natural Logarithm: Base e
Natural logarithms are an important special case where the base of the logarithm is the mathematical constant e. The value of e is approximately 2.71828 and is known as Euler's number. The natural logarithm (denoted by \(\text{ln}\)) is widely used in mathematics and engineering.
To evaluate \(\text{ln}\) of a number, like \(\text{ln} 10\) from our exercise, we use e as the base. Essentially, you are solving for the exponent that e must be raised to in order to equal the number inside the logarithm.
So, \(\text{ln} 10 \) means 'to what power must e be raised to get 10?'. Using a calculator to find this value gives us the precise number before rounding.
Rounding Decimals Correctly
Rounding decimals involves taking a number and adjusting it to a specified degree of precision. In this case, you are asked to round to four decimal places. Here鈥檚 how you do it:
  • Identify the digit in the fourth decimal place.
  • Look at the digit immediately to the right of the fourth decimal place (the fifth number).
  • If the fifth number is 5 or greater, increase the fourth decimal place by one.
  • If the fifth number is less than 5, leave the fourth decimal place as it is.
For example, if your precise calculation of \(\text{ln} 10\) gives you 2.302585, the number in the fourth decimal place is 2. The number in the fifth decimal place is 5, so we round up, resulting in 2.3026.

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Most popular questions from this chapter

Use the special properties of logarithms to evaluate each expression. \(\log _{5} 5^{6}\)

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. $$\log _{10} 36$$

Use the change-of-base rule (with either common or natural logarithms) to approximate each logarithm to four decimal places. \(\log _{\pi} 10\)

Solve each equation. Use natural logarithms. Approximate solutions to three decimal places when appropriate. $$ e^{0.012 x}=23 $$

Determine whether each statement is true or false. $$\log _{2}(64-16)=\log _{2} 64-\log _{2} 16$$
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