# Tutorial on Exponential Functions (2) - Problems

This is a tutorial on how to apply exponential functions to solve problems. Examples with detailled solutions and explanations are included.

## Example - Problem 1

The populations of 2 cities grow according to the exponential functions
P1(t) = 100e0.013 t
P2(t) = 110e0.008 t

where P1 and P2 are the populations (in thousands) of cities A and B respectively. t is the time in years such that t is positive and t = 0 corresponds to the year 2004.
When will the populations of the two cities be equal and what will be their populations?

Solution to Problem 1:

• Let t = t' be the time when P1 and P2 are equal, this leads to the following equation in t'
100 e0.013 t' = 110 e0.008 t'
• Divide both side of the above equation by 100*0.008 t'
e0.013 t' / e0.008 t' = 110/100
• Use property of exponential functions ax/ay = ax - y to rewrite the above equation as follows
e0.013 t'- 0.008 t' = 1.1
• Simplify the exponent in the left side
e0.005 t' = 1.1
• Rewrite the above in logarithmic form (or take the ln of both sides)
0.005 t' = ln 1.1
• Solve for t' and round the answer to the nearest unit.
t' = (ln 1.1) / 0.005.
t' is approximately equal to 19 years.
the year will be 2004 + 19 = 2023.
• Find the populations when t = t' = 19 years. Use any of the function P1 or P2 since they are equal at t = t'
P1(t') = 100 e0.013*19
P1(t') is approximately equal to 128 thousands.
For checking, the graphical solution to the above problem is shown below.

## Matched Problem 1

The populations of 2 cities grow according to the exponential functions
P1(t) = 120 e0.011 t

P2(t) = 125 e0.007 t

where P1 and P2 are the populations (in thousands) of cities A and B respectively. t is the time in years such that t is positive and t = 0 corresponds to the year 2004.
When will the populations of the two cities be equal and what will be their populations?

## Example Problem 2

The amount A of a radioactive substance decays according to the exponential function
A(t) = A 0 er t

where A
0 is the initial amount (at t = 0) and t is the time in days (t >= 0). Find r, assuming that the half life of this radioactive substance is 10 days.

Solution to Problem 2:

• At t = 10 days, the amount A of the substance would be equal to half the initial amount A0 (definition of half life)
A0er*10 = A0 / 2
• Divide both side of the above equation by A0
er*10 = 1 / 2
• Rewrite the above equation in logarithmic form (or take ln of both sides)
10 r = ln(1/2)
• Solve for r
r = 0.1 ln(1/2)
• Approximate r to 3 decimal places.
r is approximately equal to -0.069.
For checking, the graph of A(t) = 100 e-0.069t is shwown below. Note at t = 0 A = 100 and at t = 10 A is approximately equal to 100/2 = 50.

## Matched Problem 2

The amount A of a radioactive substance decays according to the exponential function
A(t) = A 0er t

where A
0 is the initial amount (at t = 0) and t is the time in days (t >= 0). Find r, assuming that the half life of this radioactive substance is 20 days.

## More References and Links Related to Exponential Functions

Exponential and Logarithmic Functions.
Exponential Functions.
Tutorial on Exponential and Logarithmic Equations.
Solve Exponential and Logarithmic Equations (self test).