Can someone please check these answers for me?? I'm pretty sure most of them are right, except for maybe the second T/F question...
1) T or F. The domain of y=ln x is equal to the range of y=e (superscript x).
True
2) T or F. ln(3x x y to the 3rd power) = (ln x 3) x (ln x) x (3 ln y).
True
3) Pure water has a pH of 7.0. Some acid rain has a pH of 4.5. Acid rain in how many times as acidic as pure water?
316 times more acidic
4) Suppose $3000 is invested at an annual interest rate of 4.8% compounded continously. Assume that there are no other deposits or withdrawls in this account.
What is the amount after 16 years? $6,466.35
How long would it take for the amount in the account to triple?
About 23 years
2) Ln(ab) = ln a + ln b
Ahh ok, so it's false then...thanks. :)
Let's go through each question to check the answers.
1) The statement is true. The domain of ln(x) is all positive real numbers, and the range of e^x is also all positive real numbers. Therefore, the domain of y=ln(x) is indeed equal to the range of y=e^x.
2) The statement is false. The correct property for ln(ab) is ln(a) + ln(b). So the correct expression would be ln(3x * y^3) = ln(3x) + ln(y^3) = ln(3x) + 3ln(y).
3) To determine how many times more acidic the acid rain is compared to pure water, we can calculate the difference in their pH values. pH is a logarithmic scale, so the difference in pH values represents the difference in the concentration of hydrogen ions. In this case, the pH of pure water is 7 and the pH of acid rain is 4.5. The difference is 7 - 4.5 = 2.5. Since pH is logarithmic, we can calculate the acid rain being 10^x times more acidic than pure water, where x is the difference in pH. Therefore, acid rain is 10^2.5 = 316 times more acidic than pure water.
4) For continuous compounding, we can use the formula A = P * e^(r*t), where A is the final amount, P is the initial principal, r is the annual interest rate as a decimal, and t is the time in years. Plugging in the values, we have P = $3000, r = 0.048, and t = 16. Calculating, we get A = $3000 * e^(0.048*16) = $6,466.35.
To find the time it takes for the amount to triple, we need to solve the equation A = P * e^(r*t) for t. In this case, A is 3 times the initial principal, P remains $3000, r is still 0.048, and t is the unknown time. Setting up the equation, we have 3P = P * e^(r*t). Dividing both sides by P and taking the natural logarithm, we get ln(3) = r*t. Solving for t, we have t = ln(3) / r = ln(3) / 0.048 ≈ 22.92 years, or about 23 years.
So, in summary, the answers are:
1) True
2) False
3) Acid rain is 316 times more acidic than pure water.
4) Amount after 16 years is $6,466.35
Time to triple the amount is about 23 years.