Applications of Exponential Functions

Application 1: Depreciation

Example 1

A particular sports convertible costs $\$48000$$48000. Its value depreciates ( which means to lose value) at a rate of $18%$18% per year.

$18%$18% of $48000$48000 is $8640$8640, so after $1$1 year the car will be worth $\$39360$$39360.

After the second year, the value of the car will drop by $18%$18% of $\$39360$$39360, or $\$7084.80$$7084.80, and so will be only be worth $\$32275.20$$32275.20.

 

Developing a formula

Rather than continuing to work through each year in this way, we could approach the problem differently.

If it is to drop in value by $18%$18%, then it retains $\left(1-\frac{18}{100}\right)=0.82=82%$(1−18100​)=0.82=82%.

Thus, after the first year its value will be $48000\left(1-\frac{18}{100}\right)$48000(1−18100​). After the second year, we simply apply the same factor $\left(1-\frac{18}{100}\right)$(1−18100​) to the new value. Its value would become:

$48000\left(1-\frac{82}{100}\right)\left(1-\frac{18}{100}\right)=42000\times\left(1-\frac{18}{100}\right)^2$48000(1−82100​)(1−18100​)=42000×(1−18100​)2

A pattern is emerging. After the third year the new value becomes:

$48000\times\left(1-\frac{18}{100}\right)^3$48000×(1−18100​)3

In fact the future value (often denoted $FV$FV) after $n$n years of the car is given by:

 $FV_n=48000\times\left(1-\frac{18}{100}\right)^n$FVn​=48000×(1−18100​)n

A general formula

Note that we could simplify the formula to read $FV_n=48000\left(\frac{82}{100}\right)^n$FVn​=48000(82100​)n, but it is often left as it is to remind us of how it was constructed. In general, any item, with a current value of $A$A, and depreciation rate $r%$r%, will have a future value after $n$n years given by:

$FV_n=A\left(1-\frac{r}{100}\right)^n$FVn​=A(1−r100​)n

The important principle here is that we now have a mechanism to find any future value quickly. For example, after $10$10 years the car will be worth $FV_{10}=48000\times\left(1-\frac{18}{100}\right)^{10}=6597.5$FV10​=48000×(1−18100​)10=6597.5. 

In addition logarithms can be used to determine how many years it will take for the car to depreciate down to any particular value. For example, to determine when the car will be worth half its original value, simply solve the equation $24000=48000\times\left(1-\frac{18}{100}\right)^n$24000=48000×(1−18100​)n.

$24000$24000	$=$=	$48000\times\left(1-\frac{18}{100}\right)^n$48000×(1−18100​)n
$0.5$0.5	$=$=	$\times\left(1-\frac{18}{100}\right)^n$×(1−18100​)n
$0.5$0.5	$=$=	$0.82^n$0.82n
$\log0.5$log0.5	$=$=	$n\log0.82$nlog0.82
$n$n	$\approx$≈	$3.5$3.5

So the sports car will halve in value after $3$3 years and $6$6 months.

 

Application 2: The Rule of 72

In a similar way, an amount of money invested with a bank will grow exponentially by earning bank interest.

If an amount of $\$A$$A is invested at $r%$r% for $n$n years, then the future value becomes:

$FV_n=A\left(1+\frac{r}{100}\right)^n$FVn​=A(1+r100​)n 

Note that the only difference between this formula and the previous one is the presence of a plus sign inside the brackets. The formula is developed in precisely the same way as the depreciation formula above.

Before the widespread availability of calculators, bankers used a 'rule of thumb' called the Rule of $72$72 to determine how long in years an amount invested with them took to double in value.

The rule of thumb was simply to divide the number $72$72 by the current interest rate. So for example if the bank offered $6%$6%, the number of years it took to double was approximately $\frac{72}{6}=12$726​=12 years. 

To check how accurate the rule is, using $r=6%$r=6% we find that, after $12$12 years an amount $A$A turns into:

$FV_{12}=A\left(1+\frac{6}{100}\right)^{12}=2.012A$FV12​=A(1+6100​)12=2.012A

which is slightly over the $2A$2A predicted. 

At $4%$4%, the rule of thumb says $\frac{72}{4}=18$724​=18 years to double, and the true value after $18$18 years turns out to be $2.025A$2.025A.

One way to investigate this remarkable observation is to solve for $n$n in the equation $2A=A\left(1+\frac{r}{100}\right)^n$2A=A(1+r100​)n. Thus:

$2A$2A	$=$=	$A\left(1+\frac{r}{100}\right)^n$A(1+r100​)n
$2$2	$=$=	$\left(1+\frac{r}{100}\right)^n$(1+r100​)n
$\log2$log2	$=$=	$n\log\left(1+\frac{r}{100}\right)$nlog(1+r100​)
$n$n	$=$=	$\frac{\log2}{\log\left(1+\frac{r}{100}\right)}$log2log(1+r100​)​

Hence we need to compare the values of $\frac{\log2}{\log\left(1+\frac{r}{100}\right)}$log2log(1+r100​)​ and $\frac{72}{r}$72r​ across a range of interest rates. The comparison is depicted in the graph below. Astonishingly, the rule of $72$72 is exceptionally close to the true doubling time.

A POSSIBLE EXPLANATION

The quantity $\frac{\log2}{\log\left(1+\frac{r}{100}\right)}$log2log(1+r100​)​ can be written $\frac{\ln2}{\ln\left(1+\frac{r}{100}\right)}$ln2ln(1+r100​)​ using the change of base formula. The denominator $\ln\left(1+\frac{r}{100}\right)$ln(1+r100​) can be shown mathematically to to be very close to the fraction $\frac{r}{100}$r100​ provided $r$r is reasonably small. Check this with your calculator.

Hence the quantity $\frac{\ln2}{\ln\left(1+\frac{r}{100}\right)}$ln2ln(1+r100​)​ is very slightly less than $\frac{\ln2}{\frac{r}{100}}=\frac{100\ln2}{r}$ln2r100​​=100ln2r​ or approximately $\frac{69}{r}$69r​. Then perhaps, after checking, the number $69$69 may have been lifted to $72$72 to compensate for the shortfall in the original approximation. The added advantage is that $72$72 has lots of factors!

What ever the reason, it is an excellent rule of thumb! 

 

Worked Examples

Question 1

Question 2