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Chapter 3: Problem 96

Suppose \(g(b)=\log _{b} 5,\) where the domain of \(g\) is the interval \((1, \infty)\). Is \(g\) an increasing function or a decreasing function?

Short Answer

Expert verified
The function \(g(b) = \log_{b} 5\) is a decreasing function in its domain \((1, \infty)\), as its derivative \(\frac{-\ln 5}{b (\ln b)^2 }\) is always negative for \(b > 1\).

Step by step solution

01

Find the derivative of g(b) with respect to b.

To find the derivative, we will first change the expression to the natural logarithm since it is easier to differentiate. Recall the logarithmic base change formula: \(\log_{b} 5 = \frac{\ln 5}{\ln b}\). Now, let's find the derivative of \(g(b) = \frac{\ln 5}{\ln b}\) with respect to \(b\):
02

Apply the quotient rule for differentiation

We apply the quotient rule for differentiating functions that are of the form \(\frac{u(x)}{v(x)}\), which states that: \[\frac{d}{dx} \left( \frac{u(x)}{v(x)} \right) = \frac{v(x)u'(x) - u(x)v'(x)}{[v(x)]^2}.\] In our case, we have \(u(x)= \ln 5\) and \(v(x) = \ln b\). Notice that \(\frac{d}{db} \ln 5 = 0\) because \(\ln 5\) is a constant and \(\frac{d}{db} \ln b = \frac{1}{b}\). Now we apply the quotient rule:
03

Calculate the derivative

Using the quotient rule, we have: \[\frac{d}{db}g(b)= \frac{\ln b * 0 - \ln 5 * \frac{1}{b}}{(\ln b)^2} = \frac{-\ln 5}{b (\ln b)^2 }.\]
04

Analyze the sign of the derivative

Now we examine the sign of the derivative on the given domain \((1, \infty)\). Notice that on the domain \(b>1\), \(\ln 5\) is always positive as \(5>1\). At the same time, \((\ln b)^2\) and \(b\) are also positive for \(b > 1\). Therefore, the derivative \(\frac{d}{db}g(b)\) is always negative for \(b>1\).
05

Determine if the function is increasing or decreasing

Since the derivative is negative for all values of \(b\) in the given domain \((1, \infty)\), we conclude that \(g(b) = \log_{b} 5\) is a decreasing function in its domain.

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