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Chapter 4: Problem 123

Write the domain in interval notation. $$ s(x)=\ln (\sqrt{x+5}-1) $$

Short Answer

Expert verified
( -4, \infty )

Step by step solution

01

Understand the Argument of the Logarithm

Identify the part of the function inside the logarithm, which is \( \sqrt{x+5} - 1 \). The domain of a logarithmic function is defined only when its argument is positive.
02

Set the Argument Greater Than Zero

To ensure the logarithm is defined, set \( \sqrt{x+5} - 1 > 0 \) and solve for \( x \).
03

Isolate the Square Root

Add 1 to both sides of the inequality: \( \sqrt{x+5} > 1 \).
04

Square Both Sides

Eliminate the square root by squaring both sides: \( x+5 > 1 \).
05

Solve for x

Subtract 5 from both sides: \( x > -4 \). Thus, the domain of the function must be all \( x \) values greater than \( -4 \).
06

Write in Interval Notation

The domain in interval notation is \(( -4, \infty ) \).

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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 to represent the set of all numbers between two endpoints.
You use parentheses '()' to indicate that an endpoint is not included in the interval and brackets '[]' to show that an endpoint is included.
For example, the interval \( (a, b) \) means all numbers greater than \( a \) but less than \( b \), excluding \( a \) and \( b \).
For the exercise provided, we determined the domain of the function \( s(x) = \ln(\sqrt{x+5} - 1) \) and wrote it as \( (-4, \infty) \).
This means \( x \) can be any number greater than \( -4 \) but not equal to it.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions.
The natural logarithm, denoted as \ln(x)\, is the logarithm to the base \( e \) (where \( e \) is approximately 2.718).
For a logarithmic function to be defined, its argument must be greater than zero.
The function \( s(x) = \ln(\(\sqrt{x+5} - 1\)) \) means we're looking at the logarithm of the expression \( \sqrt{x+5} - 1 \).
This expression inside the logarithm needs to be positive for the function to exist.
Therefore, we solve the inequality \( \sqrt{x+5} - 1 > 0 \) to find the domain.
Domain Restrictions
Domain restrictions tell us which values of \( x \) are allowed for a function.
For logarithmic functions, the argument inside the log must always be positive.
Let's break down the steps to determine the domain of \( s(x) = \ln(\(\sqrt{x+5} - 1\)) \):
  • First, identify the argument: \( \sqrt{x+5} - 1 \).
  • Then, set the argument greater than zero: \( \sqrt{x+5} - 1 > 0 \).
  • Add 1 to both sides: \( \sqrt{x+5} > 1 \).
  • Square both sides: \( x + 5 > 1 \).
  • Solve for \( x \): \( x > -4 \).
Thus, the domain is all \( x \) values greater than \( -4 \), written in interval notation as \( (-4, \infty) \).

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Most popular questions from this chapter

(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. $$ \log _{2} 15 $$

Determine if the given value of \(x\) is a solution to the logarithmic equation. \(\log _{4} x=3-\log _{4}(x-63)\) a. \(x=64\) b. \(x=-1\) c. \(x=32\)

Fluorodeoxyglucose is a derivative of glucose that contains the radionuclide fluorine- \(18\left({ }^{18} \mathrm{~F}\right) .\) A patient is given a sample of this material containing \(300 \mathrm{MBq}\) of \({ }^{18} \mathrm{~F}\) (a megabecquerel is a unit of radioactivity). The patient then undergoes a PET scan (positron emission tomography) to detect areas of metabolic activity indicative of cancer. After \(174 \mathrm{~min}\), one-third of the original dose remains in the body. a. Write a function of the form \(Q(t)=Q_{0} e^{-k t}\) to model the radioactivity level \(Q(t)\) of fluorine- 18 at a time \(t\) minutes after the initial dose. b. What is the half-life of \({ }^{18} \mathrm{~F}\) ?

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(1024=19^{x}+4\)

Solve the equation. Write the solution set with the exact values given in terms of common or natural logarithms. Also give approximate solutions to 4 decimal places. \(11^{1-8 x}=9^{2 x+3}\)
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