Solving Exponential Word Problems

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Know: Exponentials and Logarithms Analysis: Let t = number of half lives0.27=1.00(0.5)) Take the log of both sides (remember we are solving for t here)log (0.27)=(t)log (0.5) Use the power property to bring down tt=log(0.27)/log(0.5) Solve for tt=1.89 half life Simplify100=(1.89) (half life length) Now you know how many half lives are in 100 yrs. Half life length = 100/1.89 Simplify Half life length = 53 years Simplify Example of Continuously Compounded Interest: Express the amount of time (t) money is placed in an account with continuously compounded interest in terms of the interest rate (r), principle amount of money (P) and the desired amount of money (A)Given: Time-t Principle-PRate-r Desired amount of money-AWant:t in terms of A,r,and PKnow: A=Pe) Take the natural log of both sidesln(A/P)=rt Use the power property and simplifyt=ln(A/P)/r Divide both sides by r Example of compound interest Find the amount of money in an account with holding $200 at 5% interest compounded semiannually for 8 years.

Given: Principle amount of money P = 0Interest rate r = 0.05Number of times compounded per year n = 2Time t = 8 years Amount of money after t years AWant: A after 8 years Know: A=P (1 r / n) A=6.90 Solve for AExample of population growth Bacteria in a petri dish double in population every hour. Assuming the absence of limiting factors, how many bacteria are there after 24 hours?To solve problems on this page, you should be familiar with Suppose that the population of rabbits increases by 1.5 times a month. When the initial population is 100, what is the approximate integer population after a year?The half-life of carbon-14 is approximately 5730 years.Given:3 bacteria in a petri dish They double every hour24 hours Want: How many after 24 hours?Solving Exponential and Logarithmic Functions Key Terms o Exponential function o Logarithmic function o Natural logarithmic function Objectives o Know the basic properties of exponential and logarithmic functions o Learn how to apply these functions to solving problems, including word problems We should already know that certain expressions involve a variable raised to a power other than 1.Exponential functions tell the stories of explosive change.The two types of exponential functions are exponential growth and exponential decay.Know: Exponentials and Logarithms Analysis: Let t = number of half lives Let A = amount of isotope after t years A=200(0.5A=50 grams left Simplify Another Example: What is the half life of Isotope Y if it decays to 27% of its original amount in 100 years?Given: Decays to 27% of its original 100% after 100 years Want: What is the half life?It is essentially the same except for the fact that you will be increasing, so instead of multiplying by a number less than 1 you will have to multiply by a number greater than one.You still have to find the trend and then use that trend to solve for the given number of intervals, in this case 8 years.

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