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Chapter 4: Problem 116

Determine whether the statement is true or false. Justify your answer. The graph of \(f(x)=\log _{3} x\) contains the point (27,3).

Short Answer

Expert verified
True. The statement is correct because 3 to the power of 3 equals 27.

Step by step solution

01

Understand the logarithmic function

The function in the exercise is a logarithm with base 3. In other words, the result of the function \(f(x) = \log_{3} x\) gives us the exponent to which we need to raise 3 to get x.
02

Use the properties of logarithms

According to the properties of logarithms, if \(f(x) = \log_{b} x\), then \(b^{f(x)} = x\). Therefore, in order to check if the point (27,3) lies on the graph of the function \(f(x) = \log_{3} x\), we need to check whether raising 3 to the power of 3 equals 27.
03

Calculate 3 to the power of 3

Calculate the value of 3 to the power of 3. This gives us 27.
04

Check if the value of 3 to the power of 3 equals 27

Since 3 to the power of 3 equals 27, this confirms that the point (27,3) lies on the graph of \(f(x) = \log_{3} x\)

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Most popular questions from this chapter

What type of visual display can you create to get an idea of which type of model will best fit the data set?

Use the regression feature of a graphing utility to find a power model \(y=a x^{b}\) for the data and identify the coefficient of determination. Use the graphing utility to plot the data and graph the model in the same viewing window. $$(0.5, 1.0), (2, 12.5), (4, 33.2), (6, 65.7), (8, 98.5),(10, 150.0)$$

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