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You overhear a student talking about a property of logarithms in which division becomes subtraction. Explain what the student means by this.

By 2019 , nearly 1 dollar out of every 5 dollars spent in the U.S. economy is projected to go for health care. The bar graph shows the percentage of the U.S. gross domestic product (GDP) going toward health care from 2007 through 2014 , with a projection for 2019.(GRAPH CAN'T COPY). The data can be modeled by the function \(f(x)=1.2 \ln x+15.7\) where \(f(x)\) is the percentage of the U.S. gross domestic product going toward health care \(x\) years after \(2006 .\) Use this information to solve. a. Use the function to determine the percentage of the U.S. gross domestic product that went toward health care in \(2008 .\) Round to the nearest tenth of a percent. Does this underestimate or overestimate the percent displayed by the graph? By how much? b. According to the model, when will \(18.6 \%\) of the U.S. gross domestic product go toward health care? Round to the nearest year.

The \(p H\) 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 pH 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. (GRAPH CAN'T COPY). 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)?

Students in a psychology class took a final examination. As part of an experiment to see how much of the course content they remembered over time, they took equivalent forms of the exam in monthly intervals thereafter. The average score for the group, \(f(t),\) after \(t\) months was modeled by the function $$ f(t)=88-15 \ln (t+1), \quad 0 \leq t \leq 12 $$ a. What was the average score on the original exam? b. What was the average score after 2 months? 4 months? 6 months? 8 months? 10 months? one year? c. Sketch the graph of \(f\) (either by hand or with a graphing utility). Describe what the graph indicates in terms of the material retained by the students.

The \(p H\) 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 pH 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. (GRAPH CAN'T COPY). 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 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)?

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because logarithms are exponents, the product, quotient, and power rules remind me of properties for operations with exponents.

Explain how to solve an exponential equation when both sides cannot be written as a power of the same base. Use \(3^{x}=140\) in your explanation.

Explain why the logarithm of 1 with base \(b\) is 0.

In many states, a \(17 \%\) risk of a car accident with a blood alcohol concentration of 0.08 is the lowest level for charging a motorist with driving under the influence. Do you agree with the \(17 \%\) risk as a cutoff percentage, or do you feel that the percentage should be lower or higher? Explain your answer. What blood alcohol concentration corresponds to what you believe is an appropriate percentage?