Tutorial & Calculator Based on natural logs and the exponent "e".To see exponential growth based on common logs and base 10 exponents, click here.To read about exponential decay particularly half-life, click here.
There are four exponential growth problems further down this page.
What was the bacteria population at the beginning of the experiment (four hours ago.)?
^{(k•t)}
a = 80,000 ÷ 2.71828...
a = 80,000 ÷ 2.71828... a = 80,000 ÷ 2.22554 Beginning Amount = 35,946 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
What will be its population in 2016?
^{k*t}
y(t) = 33,476,688 × 2.71828...
y(t) = 33,476,688 × 2.71828... y(t) = 33,476,688 × 1.0839311486 36,286,425 * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
What is the interest rate of this account? For the purpose of solving this problem we'll say that: a=100 y(t)=200
k = (ln [200 ÷ 100]) ÷ 8 k = (ln [2]) ÷ 8 k = (0.6931471806) ÷ 8 k = 0.0866433976 k = 8.66433976%
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How long will it take the population to triple? (That is, when will the population be 7,500?)
first must solve for the rate.k = (ln [y(t) ÷ a]) ÷ t k = (ln [3,000 ÷ 2,500]) ÷ 2 k = (ln [1.2]) ÷ 2 k = ( 0.182321556793955 ) ÷ 2 k = 0.0911607783969773
t = (ln [y(t) ÷ a]) ÷ k t = (ln [7,500 ÷ 2,500]) ÷ 0.0911607783969773 t = (ln [3]) ÷ 0.0911607783969773 t = (1.09861228866811) ÷ 0.0911607783969773 t = 12.0513702053 hours
rate function, enter the numbers, and then use this rate calculation after we click the time function.
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