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Chapter 10: Problem 23

Find each logarithm. Give approximations to four decimal places. \(\ln 10\)

Short Answer

Expert verified
2.3026

Step by step solution

01

Use Natural Logarithm Definition

Understand that \(\text{ln}\) represents the natural logarithm, which is the logarithm to the base \(e\). To find \(\text{ln} 10\), you need to calculate the power to which \(e\) (approximately 2.71828) must be raised to obtain 10.
02

Apply Calculator

Use a scientific calculator to find the natural logarithm of 10. Enter \(10\) and press the \(\text{ln}\) button.
03

Record the Value

The result from the calculator should be approximately 2.3026. This value represents \(\text{ln} 10\) rounded to four decimal places.

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Key Concepts

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

logarithms
Logarithms are a powerful mathematical concept that allow us to solve for unknown exponents. A logarithm answers the question, 'To what power must we raise a certain base to get a specific number?'. For example, in the expression \(\text{log}_b(x) = y\),
we are asking, 'What exponent \(y\) do we need for base \(b\) to result in \(x\)?'.

Common bases for logarithms include:
  • Base 10 (common logarithm, written as \(\text{log}_{10}\) or simply \(\text{log}\))
  • Base \(e\) (natural logarithm, written as \(\text{ln}\))
These make it easier to deal with large numbers in scientific calculations.

In summary, logarithms transform multiplicative relationships into additive ones, making operations simpler.
natural logarithm
The natural logarithm (\(\text{ln}\)) is a special type of logarithm that uses the base \(e\). The constant \(e\) is approximately equal to 2.71828 and is irrational, just like \(\pi\).

Here are a few key points to understand about natural logarithms:
  • The natural logarithm of a number \(x\) tells you the power to which \(e\) must be raised to produce \(x\).
  • It is often used in science and engineering due to its unique mathematical properties, particularly in cases involving growth and decay rates such as interest calculations and population models.
  • It is denoted as \(\text{ln}(x)\).


Therefore, to find \(\text{ln}(10)\), we seek the exponent to which \(e\) must be raised to equal 10. Using a scientific calculator, the approximate value of \(\text{ln}(10)\) is found to be 2.3026.
approximations
Approximations are used in mathematics to find a value that is close enough to the correct answer, usually when the exact value is difficult or impossible to calculate.

For example, when we use a calculator to find \(\text{ln}(10)\), it gives an approximate value because the true value is an infinite decimal. Here are some points on why we use approximations:
  • They make complex calculations manageable and understandable.
  • They are practical for real-world applications where exact values are often unnecessary or impossible to obtain.
  • They allow us to use simpler numbers, especially when dealing with irrational numbers like \(e\), \(\pi\), or square roots.


Therefore, in the context of \(\text{ln}(10)\), we approximate the result to 2.3026, rounded to four decimal places. This level of precision is typically sufficient for most scientific and engineering calculations.

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Most popular questions from this chapter

Solve each equation. Use natural logarithms. When appropriate, give solutions to three decimal places. See Example 2. $$ e^{-0.103 x}=7 $$

The concentration of a drug injected into the bloodstream decreases with time. The intervals of time \(T\) when the drug should be administered are given by $$T=\frac{1}{k} \ln \frac{C_{2}}{C_{1}}$$ where \(k\) is a constant determined by the drug in use, \(C_{2}\) is the concentration at which the drug is harmful, and \(C_{1}\) is the concentration below which the drug is ineffective. (Source: Horelick, Brindell and Sinan Koont, "Applications of Calculus to Medicine: Prescribing Safe and Effective Dosage," UMAP Module 202.) Thus, if \(T=4,\) the drug should be administered every 4 hr. For a certain drug, \(k=\frac{1}{3}, C_{2}=5,\) and \(C_{1}=2 .\) How often should the drug be administered?

Solve equation. \(\log _{1 / 2} 8=x\)

Use the properties of logarithms to write each expression as a single logarithm. Assume that all variables are defined in such a way that the variable expressions are positive, and bases are positive numbers not equal to 1. $$ 3 \log _{a} 5-\frac{1}{2} \log _{a} 9 $$

Let \(m\) be the number of letters in your first name, and let \(n\) be the number of letters in your last name. (a) In your own words, explain what \(\log _{m} n\) means. (b) Use your calculator to find \(\log _{m} n .\) (c) Raise \(m\) to the power indicated by the number found in part (b). What is your result?
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