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Chapter 12: Problem 8
Solve each exponential equation by expressing each side as a power of the same base and then equating exponents. $$5^{3 x-1}=125$$
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What is a logarithmic equation?
The pH scale is used to measure the acidity or alkalinity of a solution. The scale ranges from 0 to \(14 .\) A neutral solution, such as pure water, has a pH of 7. An acid solution has a pH less than 7 and an alkaline solution has a p \(H\) greater than \(7 .\) The lower the \(p H\) below \(7,\) the more acidic is the solution. Each whole-number decrease in \(p H\) represents a tenfold increase in acidity. The \(p H\) of a solution is given by $$\mathrm{pH}=-\log x$$ where \(x\) represents the concentration of the hydrogen ions in the solution, in moles per liter. Use the formula to solve. Express answers as powers of \(10 .\) a. Normal, unpolluted rain has a pH of about 5.6. What is the hydrogen ion concentration? b. An environmental concern involves the destructive effects of acid rain. The most acidic rainfall ever had a \(\mathrm{pH}\) of \(2.4 .\) What was the hydrogen ion concentration? c. How many times greater is the hydrogen ion concentration of the acidic rainfall in part (b) than the normal rainfall in part (a)?
The loudness level of a sound, \(D,\) in decibels, is given by the formula $$D=10 \log \left(10^{12} I\right)$$ where I is the intensity of the sound, in watts per meter \(^{2} .\) Decibel levels range from \(0,\) a barely audible sound, to \(160,\) a sound resulting in a nuptured eardrum. Use the formula to solve Exercises. What is the decibel level of a normal conversation, \(3.2 \times 10^{-6}\) watt per meter \(^{2} ?\)
a. Simplify: \(e^{\ln 3}\) b. Use your simplification from part (a) to rewrite \(3^{x}\) in terms of base \(e\)
Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. When graphing a logarithmic function, I like to show the graph of its horizontal asymptote.
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