2A Calc Logarithmic Functions

What you're Solving for Key:

6. Simplyfy the expression completly

7+14 Solve for x using logs

24 solve for t assume a and b are are positve a and b =/ 1 and k is non zero (slash is supposed to be through the equal sign)

26+28 put functions in the form p=p0^e^kt

30 find the inverse

33 find k such that p=p0^e^kt

***PLEASE SHOW ALL WORK****

\f32. The exponential function y(x) = Cex satisfies the conditions y(0) = 2 and y(1) = 1. Find the constants C and a What is y(2)?\f38. The population of the US was 282.2 million in 2000 and 327.2 million in 2018. Assuming exponential growth, a. In What year is the population expected to go over 350 million? b. What population is predicted for the 2020 census? 44. A cup of coffee contains 100 mg of caffeine, which leaves the body at a continuous rate of 17% per hour. a. Write a formula for the amount, A mg, of caffeine in the body t hours after drinking a cup of coffee. b. Graph the function from part (a). Use the graph to estimate the half-life of caffeine. c. Use logarithms to find the half-life of caffeine.49. The size of an exponentially growing bacteria colony doubles in 5 hours. How long will it take for the number of bacteria to triple?The formula p = po . ext describes exponential growth, where: - p is the population at time t. - po is the initial population at time t = 0. - k is the growth constant. - t is the time. If you want the population to double in a given time, you're looking for a specific value of k that satisfies this condition. When the population doubles, it means that p = 2po, because doubling means multiplying by 2. So, we have: 2po = po . ekt Dividing both sides by po (assuming po # 0): 2 = ekt Taking the natural logarithm (In) of both sides: In(2) = In(ekt) Using the property In(e?) = x: In (2) = kt Now, solve for k: k = In (2) t This formula gives you the value of k such that the population will double in time t