What is formula for Logarithmic Growth (inverse of #P(t)=P_0e^{rt}#)?

Question

I'm looking for Logarithmic Growth formula that is inverse of Exponential Growth formula below.

Exponential Growth (Malthusian growth model):

#P(t)=P_0e^{rt}#

P(t) = the amount of some quantity at time t
P0 = initial amount at time t = 0
r = the growth rate
t = time (number of periods)

There is also a more complicated formula for Logistic Growth: https://en.wikipedia.org/wiki/Logistic_function#In_ecology:_modeling_population_growth

But what I'm looking for is a simpler formula for normal Logarithmic Growth (without saturation and limit parameters and without exponential growth at the start).

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Answer 1 · 2017-06-27T00:01:03

#P^-1(t)= (ln(t)-ln(P_0))/r#

Given: #P(t)=P_0e^{rt} " [1]"# find #P^-1(t)#:

Substitute #P^-1(t)# for t in equation [1]:

#P(P^-1(t))=P_0e^{rP^-1(t)} " [1.1]"#

By definition the left becomes t:

#t=P_0e^{rP^-1(t)} " [1.2]"#

Divide both sides by #P_0#

#t/P_0=e^{rP^-1(t)} " [1.3]"#

Use the natural logarithm on both sides:

#ln(t/P_0)=ln(e^{rP^-1(t)}) " [1.4]"#

ln and e cancel on the right:

#ln(t/P_0)=rP^-1(t) " [1.5]"#

Flip the equation and divide by r:

#P^-1(t)= ln(t/P_0)/r " [1.6]"#

Use a property of logarithms where division become subtraction:

#P^-1(t)= (ln(t)-ln(P_0))/r " [1.7]"#

Equation [1.7] is a candidate for the inverse but Inverse verification requires that #P(P^-1(t)) = t = P^-1(P(t))#:

#P(P^-1(t))=P_0e^{r((ln(t)-ln(P_0))/r)}#

#P(P^-1(t))=P_0e^{((ln(t)-ln(P_0)))}#

#P(P^-1(t))=P_0e^(ln(t/P_0))#

#P(P^-1(t))=P_0t/P_0#

#P(P^-1(t))=t#

#P^-1(P(t))= (ln(P_0e^{rt})-ln(P_0))/r#

#P^-1(P(t))= (ln(P_0e^{rt}/P_0))/r#

#P^-1(P(t))= (ln(e^{rt}))/r#

#P^-1(P(t))= rt/r#

#P^-1(P(t))= t#

Verified #P^-1(t)= (ln(t)-ln(P_0))/r " [1.7]"#