# Inverse, Exponential, and Logarithmic Functions

## Contents

#### Problems for "Exponential Growth and Decay"

Problem : If P = 300e 2t , at what time will P=600?

 600 = 300e 2t 2 = e 2t

use natural log to solve for t

 ln(2) = 2t t = .347

Problem : A strain of bacteria multiply in such a fashion that they double in number every 4 hours. Find an expression that describes this kind of growth.

A function that grows or decays by a fixed percentage over time exhibits growth of the form

 B(t) = B 0 e kt

After 4 hours, B(t) = 2B 0 , so 2B 0 = B 0 e k(4) , 2 = e 4k Use natural log to solve for k

 ln(2) = 4k k = .173 B(t) = B 0 e .173t

Problem : Find a function that meets the following:

 = 0.693y

Any function for which

 = ky

must be of the form y = e kt so y = e 0.693t satisfies the condition above. To verify this, see that

 e 0.693t = 0.693 e 0.693t

or 0.693y

Problem : The half-life of a substance is the time that it takes for the mass of that substance to decay to 50% of its original value. If a certain substance has a half-life of 30 minutes, what is an equation that describes its decay?

This is another case of exponential decay, because a certain percentage of the substance decays at regular time intervals. So, the general form is

 C(t) = C 0 e kt

At t = 30 , 1/2 of the original remains, so

 C 0 = C 0 e k(30) = e 30k ln = 30k k - 0.0231 C(t) = C 0 e -0.0231t

Problem : After 40 minutes, only 34% of a radioactive compound is remaining. What is the expression describing its decay?

 C(t) = C 0 e kt .34C 0 = C 0 e k(40) .34 = e 40k ln.34 = 40k k - 0.027 C(t) = C 0 e -0.027t