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Chapter 4: Problem 83

Write the domain in interval notation. $$ f(x)=\log (8-x) $$

Short Answer

Expert verified
The domain is \[(-\infty, 8)\].

Step by step solution

01

- Understand the Function

The function given is \(f(x) = \log(8-x)\). To determine the domain, we need to identify the values of \(x\) for which \(\log(8-x)\) is defined.
02

- Know the Logarithm Properties

The logarithmic function \(\log(8-x)\) is defined when its argument is greater than zero. Hence, \(8-x > 0\).
03

- Solve the Inequality

Solve the inequality \(8 - x > 0\). Subtract 8 from both sides to get \(-x > -8\), then multiply by \(-1\) and reverse the inequality to obtain \(x < 8\).
04

- Write the Domain in Interval Notation

The values of \(x\) that satisfy \(x < 8\) are all real numbers less than 8. In interval notation, this is written as \[(-\infty, 8)\].

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Key Concepts

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

Interval Notation
Interval notation is a way of writing the set of all numbers between two endpoints. This notation helps in expressing domains and ranges concisely. For example, if the domain of a function includes all real numbers less than 8, we use interval notation to represent it as \((-\infty, 8)\). Here, \(-\infty\) indicates that there is no lower limit to the numbers included, and 8 is the upper limit. The round bracket next to 8 indicates that 8 itself is not included in the domain.
To determine how to write the domain in interval notation, follow these steps:
  • Identify the interval's lower bound and upper bound.
  • Check if the bounds are included in the domain (closed interval using square brackets) or not included (open interval using round brackets).
  • Express the interval using appropriate symbols.
Logarithm Properties
Understanding logarithm properties is crucial when dealing with logarithmic functions. One important property is that the argument of a logarithm must be positive because you cannot take the logarithm of a negative number or zero.
For instance, in the function \(f(x) = \log(8-x)\), the argument is \(8-x\). This means we need \(8-x > 0\). This requirement arises directly from the properties of logarithms.
Important logarithm properties to remember include:
  • The argument of the logarithm must be greater than zero \(\log(x)\) is defined for \(x>0\).
  • Logarithms convert multiplication inside the log into addition outside the log: \ \log(ab) = \log(a) + \log(b) \.
  • Logarithms turn division inside the log into subtraction outside the log: \ \log\left(\frac{a}{b}\right) = \log(a) - \log(b) \.
  • Exponents inside the log can come out as coefficients: \ \log(a^b) = b \cdot \log(a) \.
Solving Inequalities
Solving inequalities is an essential skill when finding the domains of functions involving variables. The solution to an inequality gives us a range of values that satisfy a particular condition.
Consider the inequality \(8 - x > 0\) from the logarithmic function \(f(x) = \log(8-x)\). To solve it:
  • Start by isolating the variable. Here, subtracting 8 from both sides gives \(-x > -8\).
  • Next, multiply both sides by \(-1\), remembering to reverse the inequality sign, resulting in \(x < 8\).

This means that \(x\) can take any value less than 8. When translated to interval notation, this solution is written as \((-\infty, 8)\).
Key steps in solving inequalities include:
  • Isolating the variable on one side of the inequality.
  • Performing arithmetic operations systematically.
  • Reversing the inequality sign when multiplying or dividing by a negative number.
  • Expressing the final solution in interval notation.

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Most popular questions from this chapter

Solve the equation. Write the solution set with the exact solutions. Also give approximate solutions to 4 decimal places if necessary. \(\log _{7}(12-t)=\log _{7}(t+6)\)

(See Example 8 ) a. Estimate the value of the logarithm between two consecutive integers. For example, \(\log _{2} 7\) is between 2 and 3 because \(2^{2}<7<2^{3}\). b. Use the change-of-base formula and a calculator to approximate the logarithm to 4 decimal places. c. Check the result by using the related exponential form. $$ \log _{3} 15 $$

Show that \(\left(\frac{e^{x}+e^{-x}}{2}\right)^{2}-\left(\frac{e^{x}-e^{-x}}{2}\right)^{2}=1\).

a. The populations of two countries are given for January 1,2000 , and for January 1,2010 . Write a function of the form \(P(t)=P_{0} e^{k t}\) to model each population \(P(t)\) (in millions) \(t\) years after January 1, 2000.$$ \begin{array}{|l|c|c|c|} \hline & \begin{array}{c} \text { Population } \\ \text { in 2000 } \\ \text { (millions) } \end{array} & \begin{array}{c} \text { Population } \\ \text { in 2010 } \\ \text { (millions) } \end{array} & \boldsymbol{P}(t)=\boldsymbol{P}_{0} e^{k t} \\ \hline \text { Switzerland } & 7.3 & 7.8 & \\ \hline \text { Israel } & 6.7 & 7.7 & \\ \hline \end{array}$$ b. Use the models from part (a) to predict the population on January \(1,2020,\) for each country. Round to the nearest hundred thousand. c. Israel had fewer people than Switzerland in the year 2000 , yet from the result of part (b), Israel will have more people in the year \(2020 ?\) Why? d. Use the models from part (a) to predict the year during which each population will reach 10 million if this trend continues.

Determine if the given value of \(x\) is a solution to the logarithmic equation. \(\log _{2}(x-31)=5-\log _{2} x\) a. \(x=16\) b. \(x=32\) c. \(x=-1\)
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