Tutorial & Calculator Based on natural logs and the exponent "e".To see exponential growth based on common logs and base 10 exponents, 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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