Transforming exponential functions to logarithmic

Question

D(t) = 82.5 · 0.9982^t with D the number of inhabitants in millions and t the time in years. t= 0 = the year 2008

 

The above mentioned equation describing the exponential growth of the number of inhabitants can be rewritten to a form t= A·log (D) + B. Rewrite this formula and determine coefficients A and B.

 

I have been struggling on finding much information online on how to solve this and feel very lost.

 

Could someone explain to me how to transform this from beginning to end?

 

Thank you so much!

Answer 1

This is an example of exponential DECAY, not exponential growth.  You have an equation of the form

 

D(t) = c d^t

 

where

 

c = 82.5

d = 0.9982

 

Solve for t using the properties of log.

 

logD = logc + t logd

 

t = (logD - logc) / logd

 

t = (1/logd)logD - logc/logd

 

A = 1/logd

B = -logc/logd

 

Now just plug in the numbers.