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Chapter 12: Problem 20
Use properties of logarithms to expand each logarithmic expression as much as possible. Where possible, evaluate logarithmic expressions without using a calculator. $$\ln \sqrt[7]{x}$$
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Describe the power rule for logarithms and give an example.
Hurricanes are one of nature's most destructive forces. These low-pressure areas often have diameters of over 500 miles. The function \(f(x)=0.48 \ln (x+1)+27\) models the barometric air pressure, \(f(x),\) in inches of mercury, at a distance of \(x\) miles from the eye of a hurricane. Use this function to solve. Use an equation to answer this question: How far from the eye of a hurricane is the barometric air pressure 29 inches of mercury? Use the \([\text { TRACE }]\) and \([\text { ZOOM }]\) features or the intersect command of your graphing utility to verify your answer.
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. The figure indicates that lemon juice has a \(\mathrm{pH}\) of \(2.3 .\) What is the hydrogen ion concentration? b. Stomach acid has a pH that ranges from 1 to 3. What is the hydrogen ion concentration of the most acidic stomach? c. How many times greater is the hydrogen ion concentration of the acidic stomach in part (b) than the lemon juice in part (a)?
The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age ten?
Evaluate each expression without using a calculator. $$\log _{2}\left(\log _{3} 81\right)$$
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