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Chapter 3: Problem 80

Use a graphing utility and the change-of-base property to graph each function. $$y=\log _{15} x$$

Short Answer

Expert verified
The graph of the function \(y=\log_{15} x\) is an upward trending curve, crossing the x-axis at x=1 and passing through the point (15, 1).

Step by step solution

01

Apply the Change-of-Base Property

The change-of-base property of logarithms states that for any positive numbers a, b, and c such that \(a \neq 1\) and \(b \neq 1\), the logarithm base a can be computed in terms of logarithms with base b as follows: \(\log_{a}{c} = \frac{\log_{b}{c}}{\log_{b}{a}}\). Applying this property to our function, the function \(y=\log_{15} x\) will be rewritten as \(y = \frac{\log{x}}{\log{15}}\), where \(\log{x}\) is the natural logarithm (base e).
02

Graph the Function

Next, graph the function \(y = \frac{\log{x}}{\log{15}}\) using a graphing utility. Typically, this will display an upward trending curve, starting from slightly above -1 when \(x < 1\), crossing the x-axis at x=1 (since any number to the power 0 equals 1), passing through the point (15, 1) and continuing to rise.
03

Interpret the Graph

The graph will help us understand the behavior of the function \(y=\log_{15} x\). The graph helps illustrate that as x increases, so does y, but at a decreasing rate. Since the base of the logarithm (15 in this case) is greater than 1, the function increases as x moves along the positive x-axis.

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Most popular questions from this chapter

From 1970 through \(2010 .\) The data are shown again in the table. Use all five data points to solve Exercises \(70-74\). $$\begin{array}{cc}\hline \begin{array}{c}x, \text { Number of Years } \\\\\text { after } 1969 \end{array} & \begin{array}{c}y, \text { U.S. Population } \\\\\text { (millions) }\end{array} \\ \hline 1(1970) & 203.3 \\\11(1980) & 226.5 \\\21(1990) & 248.7 \\\31(2000) & 281.4 \\\41(2010) & 308.7 \end{array}$$ Use your graphing utility's logarithmic regression option to obtain a model of the form \(y=a+b \ln x\) that fits the data. How well does the correlation coefficient, \(r,\) indicate that the model fits the data?

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