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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

Decide whether each statement is true or false. $$ \log _{2}(8+32)=\log _{2} 8+\log _{2} 32 $$

Solve each equation. Use natural logarithms. When appropriate, give solutions to three decimal places. See Example 2. $$ \ln e^{0.04 x}=\sqrt{3} $$

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. $$ 3 \log _{a} 5-\frac{1}{2} \log _{a} 9 $$

To four decimal places, the values of \(\log _{10} 2\) and \(\log _{10} 9\) are $$ \log _{10} 2=0.3010 \quad \text { and } \quad \log _{10} 9=0.9542 $$ Evaluate each logarithm by applying the appropriate rule or rules from this section. $$ \log _{10} \frac{2}{9} $$

The domain of \(f(x)=a^{x}\) is \((-\infty, \infty),\) while the range is \((0, \infty) .\) Therefore, since \(g(x)=\log _{a} x\) defines the inverse of \(f,\) the domain of \(g\) is _____, while the range of \(g\) is _____.
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