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Chapter 3: Problem 94

Give the coordinates of three distinct points on the graph of the function \(f\) defined by \(f(x)=\log _{3} x\).

Short Answer

Expert verified
The three distinct points on the graph of the function \(f(x) = \log_{3}{x}\) are: 1. Point 1: \((1,0)\) 2. Point 2: \((3,1)\) 3. Point 3: \((9,2)\)

Step by step solution

01

Choose three x values

First, let's choose three different x values. We will choose simple values that will make calculations easier: 1. \(x_1 = 1\) 2. \(x_2 = 3\) 3. \(x_3 = 9\)
02

Calculate f(x) for each x value

Now, we will use the given function \(f(x) = \log_{3}{x}\) to find the corresponding y-coordinates for each chosen x value: 1. For \(x_1 = 1\), \(f(1) = \log_{3}{1}\), since \(\log_{a}{1} = 0\) for any base a, then \(\log_{3}{1} = 0\). So, the corresponding coordinate for x_1 is \((1,0)\). 2. For \(x_2 = 3\), \(f(3) = \log_{3}{3}\), since \(\log_{a}{a} = 1\) for any base a, then \(\log_{3}{3} = 1\). So, the corresponding coordinate for x_2 is \((3,1)\). 3. For \(x_3 = 9\), \(f(9) = \log_{3}{9}\), since \(3^2 = 9\), \(\log_{3}{9} = 2\). So, the corresponding coordinate for x_3 is \((9,2)\).
03

Present the coordinates

With the calculated coordinates for each x value, we have: 1. Point 1: \((1,0)\) 2. Point 2: \((3,1)\) 3. Point 3: \((9,2)\) These are the three distinct points on the graph of the function \(f(x) = \log_{3}{x}\).

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