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Chapter 3: Problem 109

Use the change-of-base formula to rewrite the logarithm as a ratio of logarithms. Then use a graphing utility to graph the ratio. $$f(x)=\log _{1 / 2} x$$

Short Answer

Expert verified
After using the change of base formula, the function \(f(x)=\log _{1 / 2} x\) can be rewritten as \(f(x) = - \frac{\log x}{\log 2}\). The graph of this function is a reflection of the graph of \(\log x\) in the x-axis, with a different scale due to the division by \(\log 2\).

Step by step solution

01

Apply the Change of Base Formula

Apply the change of base formula to rewrite the function. You can choose base 10 for simplicity: \(f(x) = \log _\frac{1}{2} x = \frac{\log {x}}{\log {(1/2)}}\)
02

Simplify the function

Here the denominator is \(\log (1 / 2)\), which is \(- \log 2\) in base 10. Therefore, the function can be simplified to \(f(x) = - \frac{\log x}{\log 2}\)
03

Graph the function

Use a graphing utility, plot the function \(f(x) = - \frac{\log x}{\log 2}\). The graph would show the typical shape of a logarithm function, but reflected in the x-axis (because of the negative sign) and with a different scale (because of the division by \(\log 2\)).

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