Question

What is the formula for Logarithmic Growth (inverse of \[p(t) = {P_0}{e^{rt}}\])?

Answer 1

Hint: Here In this question, we have to solve formula for Logarithmic growth or find the inverse of \[p(t) = {P_0}{e^{rt}}\] which is a exponential Growth formula. So clearly, we have to find \[{P^{ - 1}}\left( t \right)\] of \[p(t) = {P_0}{e^{rt}}\], by taking natural logarithm on both side and further simplification using some logarithm property and by basic arithmetic operation to get the required solution.

Complete step-by-step solution:
In Mathematics, Logarithmic growth describes a phenomenon whose size can be described as a logarithm function of some input, Exponential growth is a process that increases quantity over time t and logarithmic growth is inverse of exponential growth. formula for exponential growth is given by
\[ \Rightarrow \,\,P(t) = {P_0}{e^{rt}}\]-------(1)
Where,
\[P(t)\]= the amount of quantity at time t,
\[{P_0}\]= initial amount at time t=0,
\[r\]= the growth rate and
\[t\]=time.
Consider the question we have to find the formula of logarithmic growth or inverse i.e., \[{P^{ - 1}}\left( t \right)\] of \[p(t) = {P_0}{e^{rt}}\].
substitute \[{P^{ - 1}}\left( t \right)\] for \[t\] in equation (1), then
\[ \Rightarrow P\left( {{P^{ - 1}}(t)} \right) = {P_0}{e^{r{P^{ - 1}}\left( t \right)}}\] --------(2)
As we know, by the definition of inverse \[x \cdot {x^{ - 1}} = 1\], then equation (2) becomes
\[ \Rightarrow t = {p_0}{e^{r{P^{ - 1}}\left( t \right)}}\]
Divide both side by \[{P_0}\], we have
\[ \Rightarrow \dfrac{t}{{{P_0}}} = {e^{r{P^{ - 1}}\left( t \right)}}\] ………………(3)
Apply natural logarithm i.e.,\[\ln \] on both sides, then
\[ \Rightarrow \ln \left( {\dfrac{t}{{{P_0}}}} \right) = \ln \left( {{e^{r{P^{ - 1}}\left( t \right)}}} \right)\]
\[\ln \]and \[e\]cancel on right hand side because from the property \[\ln e = 1\], then
\[ \Rightarrow \ln \left( {\dfrac{t}{{{P_0}}}} \right) = r{P^{ - 1}}\left( t \right)\]
On rearranging, we have
\[ \Rightarrow {P^{ - 1}}(t) = \dfrac{{\ln \left( {\dfrac{t}{{{P_0}}}} \right)}}{r}\]
Use quotient logarithm property in above equation that is \[\ln \left( {\dfrac{a}{b}} \right) = \ln (a) - \ln (b)\], then numerator become
\[ \Rightarrow {P^{ - 1}}(t) = \dfrac{{\ln (t) - \ln ({P_0})}}{r}\]
Hence, the formula of logarithmic growth is \[{P^{ - 1}}(t) = \dfrac{{\ln (t) - \ln ({P_0})}}{r}\].

Note: This formula i.e., formula of logarithm growth is used to calculate population growth by using some standard logarithmic properties and functions. As we know Logarithmic growth is inverse of Exponential growth so we can find the exponential growth formula \[\left( {p(t) = {P_0}{e^{rt}}} \right)\] by using a formula of logarithm growth \[\left( {{P^{ - 1}}(t) = \dfrac{{\ln (t) - \ln ({P_0})}}{r}} \right)\] this can be verification method for the above given question.