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Chapter 10: Problem 20

Find each logarithm. Give approximations to four decimal places. \(\ln 942.6\)

Short Answer

Expert verified
The value of \(l 942.6\) is approximately 6.8493.

Step by step solution

01

Recognize the Function

Identify that the problem is asking for the natural logarithm of a number. The function is \(l x\), which is the logarithm with base e.
02

Use a Calculator

Enter the number 942.6 into a scientific calculator. Then, press the \(l\) button to compute the natural logarithm.
03

Read and Record the Answer

The calculator will give you the value of \(l 942.6\). Read the value displayed and round it to four decimal places.

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Key Concepts

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

logarithmic functions
Logarithmic functions help us understand the relationships between numbers, especially those involving exponential growth or decay. A natural logarithm, represented as \(\ln...\), is a specific type of logarithm with the base \(e\), which is approximately equal to 2.71828. The natural logarithm answers the question: *To what power must e be raised to obtain a given number?* For example, \(\ln 942.6\) tells us the power to which e must be raised to get 942.6. This function is essential in various scientific and mathematical applications because it simplifies complex multiplications into manageable additions. Knowing how to use the natural logarithm effectively can make solving exponential problems much easier.
scientific calculator usage
A scientific calculator is a powerful tool for computing values like the natural logarithm. Here鈥檚 a step-by-step on how to use it:

1. Turn on your calculator and ensure you are in the correct mode, usually standard or scientific.
2. Enter the number for which you want to find the natural logarithm. In this case, type 942.6.
3. Locate and press the \(\ln\) button. It might be labeled simply as 'ln' or be found with other logarithmic functions.
4. Your calculator will display the natural logarithm of 942.6.
5. Ensure your calculator鈥檚 settings are set to provide enough decimal places, ideally four, to match the exercise requirements.

Scientific calculators often come with an instruction manual that can help you understand all available functions. Make sure to refer to it for detailed guidance.
rounding decimals
Rounding decimals is an essential skill to ensure precision in calculations while maintaining simplicity. To round decimals to a specific number of places, follow these steps:

1. Identify which decimal place you need to round to. In this exercise, it鈥檚 four decimal places.
2. Look at the digit immediately to the right of the fourth decimal place.
3. If this digit is 5 or higher, round the fourth decimal place up by one.
4. If this digit is 4 or lower, leave the fourth decimal place as it is.

For example, the computed value of \(\ln 942.6\) might be 6.849713. To round this to four decimal places:

1. Start with 6.8497.
2. The next digit is 1, which is less than 5.
3. So, the rounded value remains 6.8497.

Mastering this technique ensures that your work is both accurate and neat.

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Most popular questions from this chapter

Use the properties of logarithms to express each logarithm as a sum or difference of logarithms, or as a single number if possible. Assume that all variables represent positive real numbers. $$ \log _{4} \frac{\sqrt[4]{z} \cdot \sqrt[5]{w}}{s^{2}} $$

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. $$ 3 \log _{a} 5-4 \log _{a} 3 $$

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. $$ \log _{10} 9^{5} $$

The age in years of a female blue whale of length \(L\) in feet is approximated by \(t=-2.57 \ln \left(\frac{87-L}{63}\right)\) (a) How old is a female blue whale that measures \(80 \mathrm{ft} ?\) (b) The equation that defines \(t\) has domain \(24

Solve each problem. See Example 9. The total volume in millions of tons of materials recovered from municipal solid waste collections in the United States during the period \(1980-2007\) can be approximated by the function defined by $$ f(x)=15.94 e^{0.0656 x} $$ where \(x=0\) corresponds to \(1980, x=1\) to \(1981,\) and so on. Approximate, to the nearest tenth, the volume recovered each year. (Source: U.S. Environmental Protection Agency.) (a) 1980 (b) 1990 (c) 2000 (d) 2007
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