Activity 6.4 #1: Solving Exponential and Logarithmic Equations
Activity Design:
Activity Overview:
Individual or Group
Solve exponential equations using logarithms and interpret the real-world meaning of
the results.
Advance Preparation: None
Two of the biggest home improvement centers in the United States are Home Depot and Lowe’s. Both stores
are experiencing rapid growth, in terms of the number of stores each chain has across the country, which can
be modeled using exponential functions. Consider the following data (Source: www.homedepot.com and
www.lowes.com).
1995 1996 1997 1998 1999 2000 2001 2002 2003 2004
YEAR
Number of Home Depot Stores 423 512 624 761 930 1123 1319 1532 1707 1890
365 402 446 484 576 650 714 828 950 1087
Number of Lowe’s Stores
1. Assuming that the number of stores for each company is growing exponentially, find the average common
ratio (growth factor) for each store. What does this number tell you about the number of Home Depot and
Lowe’s Stores?
Students may approach this problem two different ways. They may compute the ratio between
successive data values and take an average of these ratios, or they may use the regression feature of
their calculator to find the exponential regression model. Either way, the intent of the question is for
students to identify the growth factor and explain what this value means.
For Home Depot, the average common ratio (growth factor) comes out 1.18 when computed by hand
(successive ratios) and 1.19 from the regression model. This number tells us that the number of Home
Depot stores is increasing by 18% or 19% each year.
For Lowe’s, the average common ratio (growth factor) comes out 1.13 when computed by hand
(successive ratios) and 1.13 from the regression model. This number tells us that the number of
Lowe’s stores is increasing by 13% each year.
2. Using your results from #1, write an exponential equation that models the data for each store. Let t = 0 be
the year 1995.
If students computed the average common ratio (growth factor) by hand (successive ratios), they
would get the following equations (models) for the number of stores. They should indicate how they
dealt with the variable T . For these results, T = 0 corresponds to 1995.
Home Depot: H (=
T ) 423 ⋅ (1.18)T
Lowe’s:
L(T
=
) 365 ⋅ (1.13)T
If students used the regression feature of their graphing calculator, the following equations (models)
would be found (if T = 0 corresponds to 1995).
Home Depot: H (=
T ) 446 ⋅ (1.19)T
Lowe’s:
L(T=
) 351 ⋅ (1.13)T
3. According to your models, will the number of Lowe’s Stores ever catch up to the number of Home Depot
stores? Justify your answer using complete sentences.
Since the rate of growth for Home Depot (18% per year) is greater than that of Lowe’s (13% per
year) and since Home Depot has more stores initially (in 1995) than Lowe’s, Lowe’s will never catch
up, according to the mathematical models. This can be seen graphically as well.
4. Home Depot has said that they will slow down the building of new stores and would be satisfied to have
2000 stores nationwide. According to the model, found earlier, when will the number of Home Depot
stores reach 2000? Write and solve an equation that answers this question. Show all your work, including
the use of logarithms in the solution process. Write a concluding sentence describing the results of your
work.
Using the model computed by hand we get
 2000 
T
ln 
 = ln (1.18 )
T
 423 
2000
= 423 ⋅ (1.18 )
 2000 
2000
ln 

= 1.18T
 423  ≈ 9.4
423
=
T
ln (1.18 )
Since the end of 1995 corresponds to T = 0 here, the result T = 9.4 tells us that in approximately
2005, Home Depot will have 2000 stores.
5. Lowe’s hopes to match the growth of Home Depot. Assuming that the growth continues as described by
your model, when will the number of Lowe’s stores reach 2000? Set up and solve, using any method you
like (symbolically, graphically, or numerically), the equation that would answer this question. Write a
concluding sentence describing the results of your work.
Using the model computed by hand we get
 2000 
T
ln 
 = ln (1.13 )
T
 365 
2000
= 365 ⋅ (1.13)
 2000 
2000
ln 
= 1.13T

365 

365
=
T
≈ 13.9
ln (1.13)
In about 2009 (about 14 years after 1995), Lowe’s will also have 2000 stores. According to the
problem situation, Lowe’s will catch up to Home Depot about 4 years after Home Depot reaches
2000 stores.