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Chapter 3: Problem 82

Use a graphing utility and the change-of-base property to graph each function. $$y=\log _{3}(x-2)$$

Short Answer

Expert verified
Using the change-of-base property, the function \(y = \log_3(x-2)\) was transformed to \(y = \frac{\ln(x-2)}{\ln(3)}\), which was then entered into a graphing utility to produce the graph of the function.

Step by step solution

01

Understand the Change-of-Base Property

The change of base formula is a mathematical property that allows us to switch the base of a log to a different number. This is especially useful if our calculator only handles log base 10 or log base e (natural log), as is the case with many basic calculators. The change-of-base property states that for any positive numbers a, b, and c (where a 鈮� 1, b 鈮� 1, and b > 0), the logarithm base a of b can be determined by the formula: \(log_a(b) = \frac{log_c(b)}{log_c(a)}\).
02

Apply the Change-of-Base Property

We apply the change-of-base property to the given function, \(y = \log_3(x-2)\), so it can easily be used with a graphing utility that uses logarithms base 10 or base e (natural log). Using base e (natural log) as an example: \(y = \log_3(x-2)\) is equivalent to \(y = \frac{\ln(x-2)}{\ln(3)}\) according to the change-of-base property.
03

Graph the function using a graphing utility

Using the transformed function \(y = \frac{\ln(x-2)}{\ln(3)}\), enter this into the graphing utility of choice and produce the graph. Ensure that the graphing utility is correctly set up to display logarithmic functions properly, adjusting the viewing window as necessary to capture the important behaviors of the graph.

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Most popular questions from this chapter

Begin by graphing \(y=|x| .\) Then use this graph to obtain the graph of \(y=|x-2|+1 . \quad \text { (Section } 1.6, \text { Example } 3)\)

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

This will help you prepare for the material covered in the first section of the next chapter. $$\text { Simplify: } \quad-\frac{\pi}{12}+2 \pi$$

Will help you prepare for the material covered in the next section. $$\text { Solve: }(x-3)^{2} > 0$$

In Exercises \(125-132,\) use your graphing utility to graph each side of the equation in the same viewing rectangle. Then use the \(x\) -coordinate of the intersection point to find the equation's solution set. Verify this value by direct substitution into the equation. $$\log _{3}(4 x-7)=2$$
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