/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 110 Use the change-of-base formula t... [FREE SOLUTION] | 91影视

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Chapter 3: Problem 110

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 4} x$$

Short Answer

Expert verified
The logarithmic function \(f(x) = \log_{1/4}{x}\) can be rewritten as \(f(x) = \frac{\log_{10}{x}}{\log_{10}{1/4}}\) or \(f(x) = \frac{\ln{x}}{\ln{1/4}}\) using the change-of-base formula. These reformulated expressions can be graphed using a graphing utility, producing the same curve as the original function.

Step by step solution

01

Understand the Exercise

The given function is \(f(x)=\log_{1/4}{x}\). The goal is to express this in terms of simpler logarithms which further could be graphed easily. The change-of-base formula will be applied to accomplish this.
02

Apply the Change-of-Base Formula

Applying the change-of-base formula: \(f(x) = \log_{1/4}{x} = \frac{\log_{10}{x}}{\log_{10}{1/4}}\) or in terms of natural logarithms \(f(x) = \frac{\ln{x}}{\ln{1/4}}\)
03

Graph the Function

Using a graphing utility, the function f(x) = \(\frac{\log_{10}{x}}{\log_{10}{1/4}}\) or f(x) = \(\frac{\ln{x}}{\ln{1/4}}\) can now be graphed.

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Most popular questions from this chapter

The populations \(P\) (in thousands) of Reno, Nevada from 2000 through 2007 can be modeled by \(P=346.8 e^{k t},\) where \(t\) represents the year, with \(t=0\) corresponding to \(2000 .\) In \(2005,\) the population of Reno was about 395,000 . (Source: U.S. Census Bureau) (a) Find the value of \(k\). Is the population increasing or decreasing? Explain. (b) Use the model to find the populations of Reno in 2010 and 2015 . Are the results reasonable? Explain. (c) According to the model, during what year will the population reach \(500,000 ?\)

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