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Chapter 3: Problem 27

Approximate the point of intersection of the graphs of \(f\) and \(g\). Then solve the equation \(f(x)=g(x)\) algebraically to verify your approximation. $$ \begin{array}{l} f(x)=\log _{3} x \\ g(x)=2 \end{array} $$

Short Answer

Expert verified
The graphs of \(f(x)=\log _{3} x\) and \(g(x)=2\) intersect at only one point when \(x = 9\).

Step by step solution

01

Set the functions equal to each other

To find the point of intersection of the two functions, set \(f(x)\) equal to \(g(x)\). This gives us the equation \(\log _{3} x = 2\).
02

Convert logarithmic to exponential form

To solve for \(x\), the equation \(\log _{3} x = 2\) should be transformed from logarithmic to exponential form. Using the definition of logarithm \(a = log_b c\) iff \(b^a = c\), the equation turns into \(3^2 = x\).
03

Calculate the value of x

Calculate the expression on the left side. Therefore, \(x = 3^2 = 9\).

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Key Concepts

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

Logarithm to Exponential Form
Understanding how to convert logarithmic to exponential form is a crucial skill when solving logarithmic equations. A logarithmic equation such as \(\log_{b} x = a\) represents the power to which the base \(b\) must be raised to produce the number \(x\). When we convert this to exponential form, the base \(b\) will be raised to the exponent \(a\) to get \(x\), which is expressed as \(b^{a} = x\).

For example, if we start with the logarithmic equation \(\log_{3} x = 2\), the exponential form would be \(3^2 = x\). Here, the base \(3\) raised to the power of \(2\) equals \(x\), which simplifies to \(x = 9\). This step is vital as it translates the logarithmic statement into an arithmetic operation that can easily be calculated.
Intersection of Graphs
The concept of graph intersection is fundamental in understanding the solutions of equations visually. The intersection point of two graphs represents the values of \(x\) and \(y\) where the two functions have the same output. To find this intersection, we typically set the equations of the two functions equal to each other, like \(f(x) = g(x)\).

For functions \(f(x) = \log_{3} x\) and \(g(x)=2\), finding their intersection implies determining the value of \(x\) for which the logarithmic function \(f\) outputs the same result as the constant function \(g\), which is \(2\) in this case. The graphical intersection point provides a visual approximation of the solution, while the algebraic method outlined in the steps confirms the exact value.
Logarithmic Functions
Logarithmic functions are the inverse of exponential functions and they have a wide range of applications in different fields such as science, engineering, and finance. A function in the form of \(y = \log_{b} x\) maps the input \(x\), a positive real number, to the power \(y\) to which the base \(b\) must be raised to obtain \(x\). The base \(b\) is a positive real number not equal to \(1\).

A key characteristic of logarithmic functions is that they increase slowly as \(x\) grows large, which makes them useful for describing quantities that grow or decay exponentially, such as population growth or radioactive decay. In the context of the given problem, the logarithmic function \(f(x) = \log_{3} x\) describes the power needed to raise \(3\) to get \(x\), and finding where this function intersects with \(g(x) = 2\) tells us the value of \(x\) when the logarithmic function outputs \(2\).

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Most popular questions from this chapter

At 8: 30 A.M., a coroner was called to the home of a person who had died during the night. In order to estimate the time of death, the coroner took the person's temperature twice. At 9: 00 A.M. the temperature was \(85.7^{\circ} \mathrm{F}\), and at 11: 00 A.M. the temperature was \(82.8^{\circ} \mathrm{F}\). From these two temperatures, the coroner was able to determine that the time elapsed since death and the body temperature were related by the formula $$t=-10 \ln \frac{T-70}{98.6-70}$$ where \(t\) is the time in hours elapsed since the person died and \(T\) is the temperature (in degrees Fahrenheit) of the person's body. (This formula is derived from a general cooling principle called Newton's Law of Cooling. It uses the assumptions that the person had a normal body temperature of \(98.6^{\circ} \mathrm{F}\) at death, and that the room temperature was a constant \(70^{\circ} \mathrm{F}\).) Use the formula to estimate the time of death of the person.

The management at a plastics factory has found that the maximum number of units a worker can produce in a day is \(30 .\) The learning curve for the number \(N\) of units produced per day after a new employee has worked \(t\) days is modeled by \(N=30\left(1-e^{k t}\right) .\) After 20 days on the job, a new employee produces 19 units. (a) Find the learning curve for this employee (first, find the value of \(k\) ). (b) How many days should pass before this employee is producing 25 units per day?

COMPARING MODELS If $$\$ 1$$ is invested in an account over a 10 -year period, the amount in the account, where \(t\) represents the time in years, is given by \(A=1+0.06 \llbracket t \rrbracket\) or \(A=[1+(0.055 / 365)]^{[365 t]}\) depending on whether the account pays simple interest at \(6 \%\) or compound interest at \(5 \frac{1}{2} \%\) compounded daily. Use a graphing utility to graph each function in the same viewing window. Which grows at a higher rate?

In a group project in learning theory, a mathematical model for the proportion \(P\) of correct responses after \(n\) trials was found to be \(P=0.83 /\left(1+e^{-0.2 n}\right)\) (a) Use a graphing utility to graph the function. (b) Use the graph to determine any horizontal asymptotes of the graph of the function. Interpret the meaning of the upper asymptote in the context of this problem. (c) After how many trials will \(60 \%\) of the responses be correct?

The demand equation for a hand-held electronic organizer is \(p=5000\left(1-\frac{4}{4+e^{-0.002 x}}\right)\) Find the demand \(x\) for a price of (a) \(p=\$ 600\) and (b) \(p=\$ 400\).
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