Applications of Logarithmic Functions 
 Section 4.4  
Annual Growth Rate 

An investment is growing continuously for t
years. P = current value, P_{0} = original amount 
A $10,000 stock investment in 1993 was worth $22,400 in 2000. Find the average annual growth rate of the stock. 
Solve 
Step 
Calculator Help 
Annual Growth Rate  Find your ln key. (Natural log)  
Identify
each part

HINT: use these key strokes  
P = 22,400 
current amount = 22,400  1 ÷ 7 × ln (2.24) [enter or =] 
P_{o}= 10,000

initial amount = 10,000  Some calculators do not require the parentheses 
t = 7

t = 2000  1993 = 7  
Annual Growth Rate  

A $5,000 stock investment in 1992 was worth $32,500 in 1998. Find the average annual growth rate of the stock. 
[Solution] 

If $10,000 is invested each year in an annuity earning 10% annual interest, when will the account be worth $1,000,000? 
*Annuities: 
P is amount invested each year. r is the annual interest rate. A is the amount after n years. 
Solve 
Step 
Calculator Help 
annuity formula  Find your log key. (common log)  
Identify
each part

HINT: use these key strokes  
P_{ }=
10,000

invested each year  log(1000000×.1÷10000 + 1)÷log(1 + .1) [enter or =] 
r = 0.1

annual interest rate  parentheses are need here 
A =
1,000,000

amount after n years  
plug in the parts  
and simplify  
Interpret Þ It will take about 25 years to have 1,000,000 by investing 10,000 a year at 10% annual interest.  
*An annuity is an invested or loan with a fixed payment and interest rate. There can be variations.  
If $5,000 is invested each year in an annuity earning 8% annual interest, when will the account be worth $500,000? 
[Solution] 

Tutorials and Applets by
Joe McDonald
Community College of Southern Nevada