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Chapter 12: Problem 5
Write each equation in its equivalent exponential form. $$5=\log _{b} 32$$
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Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I estimate that \(\log _{8} 16\) lies between 1 and 2 because \(8^{1}=8\) and \(8^{2}=64\)
Solve each equation. $$5^{x^{2}-12}=25^{2 x}$$
Use a graphing utility and the change-of-base property to graph \(y=\log _{3} x, y=\log _{25} x,\) and \(y=\log _{100} x\) in the same viewing rectangle. a. Which graph is on the top in the interval \((0,1) ?\) Which is on the bottom? b. Which graph is on the top in the interval \((1, \infty) ?\) Which is on the bottom? c. Generalize by writing a statement about which graph is on top, which is on the bottom, and in which intervals, using \(y=\log _{b} x\) where \(b>1\)
$$\text { Divide and simplify: } \frac{\sqrt[3]{40 x^{2} y^{6}}}{\sqrt[3]{5 x y}}$$
Explain how to solve an exponential equation when both sides can be written as a power of the same base.
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