Solving Exponential Equations

Exponential Functions

India is the second most populous country in the world with a population in 2008 of about 1.14 billion people. The population is growing by about 1.34% each year. We might ask if we can find a formula to model the population, P, as a function of time, t, in years after 2008, if the population continues to grow at this rate.

In linear growth, we had a constant rate of change -- a constant number that the output increased for each increase in input. For example, in the equation $f(x)=3x+4$, the slope tells us the output increases by three each time the input increases by one. This population scenario is different -- we have a percent rate of change rather than a constant number of people as our rate of change. To see the significance of this difference consider these two companies:

Company A has 100 stores, and expands by opening 50 new stores a year

Company B has 100 stores, and expands by increasing the number of stores by 50% of their total each year.

Looking at a few years of growth for these companies:

Notice that with the percent growth, each year the company is growing by 50% of the current year's total, so as the company grows larger, the number of stores added in a year grows as well.

To try to simplify the calculations, notice that after 1 year the number of stores for company B was:

$100+0.50(100)\;\;\;\;\;$or equivalently by factoring

$100(1+0.50)=150$

We can think of this as "the new number of stores is the original 100% plus another 50%."

After 2 years, the number of stores was:

$150+0.50(150)\;\;\;\;\;$or equivalently by factoring

$150(1+0.50)\;\;\;\;\;$now recall the 225 came from $100(1+0.50)^2$. Substituting that, $100(1+0.50)^2(1+0.50)$=$100(1+0.50)^3=336.5$

From this, we can generalize, noticing that to show a 50% increase, each year we multiply by a factor of (1+0.50), so after n years, our equation would be $B(n)=100(1+0.50)^n$

In this equation, the 100 represented the initial quantity, and the 0.50 was the percent growth rate. Generalizing further, we arrive at the general form of exponential functions.

Exponential Function

An exponential growth or decay function is a function that grows or shrinks at a constant percent growth rate. The equation can be written in the form

$f(x)=a(1+r)^x\;\;\;\;\;$or$\;\;\;\;\;f(x)=ab^x\;\;\;$where $b=1+r$

Where:

$a$ is the initial or starting value of the function

$r$ is the percent growth or decay rate, written as a decimal

$b$ is the growth factor or growth multiplier. Since powers of negative numbers behave strangely, we limit $b$ to positive values.

Difference between Exponential and Linear growth

To see more clearly the difference between exponential and linear growth, compare the two tables and graphs below, which illustrate the growth of the company A and B described above over a longer time frame if the growth patterns were to continue

Here is an example

Write an exponential function for India's population, and use it to predict the population in 2020.

At the beginning of the chapter we were given India's population of 1.14 billion in the year 2008 and a percent growth rate of 1.34%. Using 2008 as our starting time (t=0), our initial population will be 1.14 billion. Since the percent growth rate was 1.34%, out value for $r$ is 0.0134.

Using the basic formula for exponential growth $f(x)=a(1+r)^x$ we can write the formula, $f(t)=1.14(1+0.0134)^t$

To estimate the population in 2020, we evaluate the function at t=12, since 2020 is 12 years after 2008.

$f(12)=1.14(1+0.0134)^{12}\approx1.337$ billion people in 2020.

Have another example

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A certificate of deposit (CD) is a type of savings account offered by banks, typically offering a higher interest rate in return for a fixed length of time you will leave your money invested. If a bank offers a 24 month CD with an annual interest rate of 1.2% compound monthly, how much will a $\$1000$ investment grow to over those 24 months?

First we must notice that the interest rate is an annual rate, but is compounded monthly, meaning interest is calculated and added to the account monthly. To find the monthly interest rate, we divide the annual rate of 1.2% by 12 since there are 12 months in a year: . Each month we will earn 0.1% interest. From this, we can set up an exponential function with our initial amount of $\$1000$ and a growth rate of r=0.001, and our input m measured in months.

$f(m)=1000\begin{pmatrix}1+\dfrac{.012}{12}\end{pmatrix}^m$

$f(m)=1000(1+0.001)^m$

After 24 months, the account will have grown to $f(24)=1000(1+0.001)^24=\$1024.28$

In all the preceding examples, we saw exponential growth. Exponential functions can also be used to model quantities that are decreasing at a constant percent rate. An example of this is radioactive decay, a process in which radioactive isotopes of certain atoms transform to an atom of a different type, causing a percentage decrease of the original material over time. To see an example of this, continue to the next page.

Bismuth-210 is an isotope that radioactively decays by about 13% each day, meaning 13% of the remaining Bismuth-210 transforms into another atom (polonium=210 in this case) each day. If you begin with 100 mg of Bismuth-210, how much remains after one week?

With radioactive decay, instead of the quantity increasing at a percent rate, the quantity is decreasing at a percent rate. Our initial quantity is $a=100$ mg, and our growth rate will be negative 13%, since we are decreasing: $r=-0.13$. This gives the equation: $Q(d)=100(1-0.13)^d=100(0.87)^d$. This can also be explained by recognizing that if 13% decays, then 87% remains.

After one week, 7 days, the quantity remaining would be $Q(7)=100(0.87)^7=37.73$ mg of Bismuth-210 remains.

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$T(q)$ represents the total number of Android smart phone contracts, in thousands, held by a certain Verizon store region measured quarterly since January 1, 2012, Interpret all of the parts of the equation $T(2)=86(1.64)^2=231.3056$.

Interpreting this from the basic exponential form, we know that 86 is our initial value. This means that on Jan. 1, 2010 this region had 86,000 Android smart phone contracts. Since $b=1+r=1.64$, we know that every quarter the number of smart phone contracts grows by 64%. $T(2)=231.3056$ means that in the $2^{nd}$ quarter (or at the end of the second quarter) there were approximately 231,305 Android Smart Phone contracts.

Finding Equations of Exponential Function

In the previous examples, we were able to write equations for exponential functions since we knew the initial quantity and the growth rate. If we do not know the growth rate, but instead know only some input and output pairs of values, we can still construct an exponential function.

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In 2002, 80 deer were reintroduced into a wildlife refuge area from which the population had previously been hunted to elimination. By 2008, the population had grown to 180 deer. If this population grows exponentially, find a formula for the function.

By defining our input variable to be $t$, years after 2002, the information listed can be written as two input output pairs: (0,80) and (6,180). Notice that by choosing our input variable to be measured as years after the first year value provided, we have effectively "given" ourselves the initial value for the function: $a=80$. This gives us an equation of the form

$f(t)=80b^t$

Substituting in our second input-output pair allows us to solve for $b$:

$180=80b^6\;\;\;\;\;$Divide by 80

$b^6=\dfrac{180}{80}=\dfrac94\;\;\;\;\;$Take the $6^{th}$ root of both sides.

$b=\sqrt[6]94=1.1447$

This gives us our equation for the population:

$f(t)=80(1.1447)^t$

Recall that since $b=1+r$, we can interpret this to mean that the population growth rate is $r=0.1447$, and so the population is growing by about $14.47$% each year.

In this example, you could also have used $\begin{pmatrix}\dfrac94\end{pmatrix}^{\frac16}$ to evaluate the $6^{th}$ root if your calculator does not have a n $n^{th}$ root button.

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In the previous example, we chose to use the $f(x)=ab^x$ form of the exponential function rather than the $f(x)=a(1+r)^x$ form. This choice was entirely arbitrary -- either form would be fine to use.

When finding equations, the value for $b$ or $r$ will have to be rounded to be written easily. To preserve accuracy, it is important to not over-round these values. Typically, you want to be sure to preserve at least 3 significant digits in the growth rate. For example, If your value for $b$ was 1.00317643, you would want to round this no further than 1.00318

In the previous example, we were able to "give" ourselves the initial value by clever definition of our input variable. Next we consider a situation where we can't do this.

Find a formula for an exponential function passing through the points (-2,6) and (2,1).

Since we don;t have the initial value, we will take a general approach that will work for any function form with unkown parameters: we will substitute in both given input-output pairs in the function form $f(x)=av^x$ and solve for the unkown values, $a$ and $b$.

Substituting in (-2, 6) gives $6=ab^{-2}$

Substituting in (2,1) gives $1=ab^2$

We now solve these as a system of equations. To do so, we could try a substitution approach, solving one equation for a variable, then substituting that expression into the second equation. Solving $6=ab^{-2}$ for $a$:

$a=\dfrac6{b^{-2}}=6b^2$

In the second equation, $1=ab^2$, we substitute the expression above for $a$:

$1=(6b^2)b^2$

$1=6b^4$

$\dfrac16=b^4$

$b=\sqrt[4]{\dfrac16}\approx0.6389$

Going back to the equation a=6b^2 let us find $a$:

$a=6b^2=6(0.6389)^2=2.4492$

Putting this together gives the equation $f(x)=2.4492(0.6389)^x$

If you want another example, continue to the example on the next page

Th initial value for the function is not clear in this graph, so we will instead work using two clearer points. There are three fairly clear points: (-1.1), (1,2), and (3,4). As we saw in the last example, two points are sufficient to find the equation for a standard exponential, so we will use the latter two points.

Substituting in (1,2) gives $2=ab^1$

Substituting in (3,4) gives $4=ab^3$

Solving the first equation for $a$ gives $a=\dfrac2b$

Substituting this expression for $a$ into the second equation:

$4=ab^3$

$4=\dfrac2b b^3=\dfrac{2b^3}b\;\;\;\;\;$Simplify the right-hand side

$4=2b^2$

$2=b^2$

$b=\pm\sqrt2$

Since we restrict ourselves to positive values of $b$, we will use $b=\sqrt2$. We can then go back and find $a$:

$a=\dfrac2b=\dfrac2{\sqrt2}=\sqrt2$

This gives us a final equation of $f(x)=\sqrt2(\sqrt2)^x$

Compound Interest

In the bank certificate of deposit (CD) example earlier in the section, we encountered compound interest. typically bank accounts and other savings instruments in which earnings are reinvested, such as mutual funds and retirement accounts, utilize compound interest. The term compounding comes from the behavior that interest is earned not on the original value, but on the accumulated value of the account.

In the example from earlier, the interest was compounded monthly, so we took the annual interest rate, usually called the nominal rate or annual percentage rate (APR) and divided by 12, the number of compounds in a year, to find the monthly interest. The exponent was then measured in months.

Generalizing this, we can form a general formula for compound interest. If the APR is written in decimal form as $r$, and there are $k$ fompounding periods per year, then the interest per compounding period will be $\frac rk$. Likewise, if we are interested in the value after $t$ years, then there will be $kt$ compounding periods in that time.

Compound Interest Formula

Compound interest can be calculated using the formula

$A(t)=a\begin{pmatrix}a+\dfrac rk\end{pmatrix}^{kt}$

Where:

$A(t)$ is the account value

$t$ is measured in years

$a$ is the starting amount of the account, often called the principal

$r$ is the annual percentage rate (APR), also called the nominal rate

$k$ is the number of compounding periods in one year

If you invest $\$3,00$ in an investment account paying 3% interest compounded quarterly, how much will the account be worth in 10 years?

Since we are starting with $\$3,00$, $a=3000$ Our interest rate is 3%, so $r=0.03$ Since we are compounding quarterly, we are compounding 4 times per year, so $k=4$ We want to know the value of the account in 10 years, so we are looking for $A(10)$, the value when $t=10$.

$A(10)=300\begin{pmatrix}1+\dfrac{0.03}4\end{pmatrix}^{4(10)}=\$4045.05$

The account will be worth $\$4045.05$ in 10 years.

If you want another example, continue to the next page

A 529 plan is a college savings plan in which a relative acan invest money to pay for a child's later college tuition, and the account grows tax free. If Lily wants to set up a 529 account for her new granddaughter, wants the account to grow to $\$40,000$ over 18 years, and she believes the account will earn 6% compounded semi-annually (twice a year), how much will Lily need to invest in the account now?

Since the account is earning 6%, $r=0.06$

Since interest is compounded twice a year, $k=2$

In this problem, we don't know how much we are starting with, so we will be solving for $a$, the initial amount needed. We do know we want the end amount to be $\$40,000$, so we will be looking for the value of $a$ so that $A(18)=40,000$.

$40,000=A(18)=a\begin{pmatrix}1+\dfrac{0.06}2\end{pmatrix}^{2(18)}$

$40,000=a(2.8983)$

$a=\dfrac{40,000}{2.8983}\approx\$13,801$

Lily will need to invest $\$13,801$ to have $\$40,000$ in 18 years.

Because of compounding throughout the year, with compound interest the actual increase in a year is more than the annual percentage rate. If $\$1,00$ were invested at 10%, the table below shows the value after 1 year at different compounding frequencies:

If we were to compute the actual percentage increase for the daily compounding, there was an increase of $\$105.16$ from an original amount of $\$1,000$, for a percentage increase of $\dfrac{105.16}{1000}=0.10516=10.516$ increase. This quantity is called the annual percentage yield (APY).

Notice that given any starting amount, the amount after 1 year would be $A(1)=a\begin{pmatrix}1+\dfrac rk\end{pmatrix}^k$. To find the total change, we would subtract the original amount, then to find the percentage change we would divide that by the original amount:

$\dfrac{a\begin{pmatrix}1+\dfrac rk\end{pmatrix}^k-a}a=\begin{pmatrix}1+\dfrac rk\end{pmatrix}^k-1$

Annual Percentage Yield

The annual percentage yield is the actual percent a quantity increases in one year. It can be calculated as $APY=\begin{pmatrix}1+\dfrac rk\end{pmatrix}^k-1$

Notice this is equivalent to finding the value of $\$1$ after 1 year, and subtracting the original dollar.

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Bank A offers an account paying 1.2% compounded quarterly. Bank B offers an account paying 1.1% compounded monthly. Which is offering a better rate?

We can compare these rates using the annual percentage yield - the actual percent increase in a year.

Bank A: $APY=\begin{pmatrix}1+\dfrac{0.012}4\end{pmatrix}^4 -1 =0.01254=1.2054$

Bank B: $APY =\begin{pmatrix}1+\dfrac{0.011}{12}\end{pmatrix}^12-1=0.011056=1.1056$

Bank B's monthly compounding is not enough to catch up with Bank A's better APR. Bank A offers a better rate.

A Limit to Compounding

As we saw earlier, the amount we earn increases as we increase the compounding frequency. The table, though, shows that the increase from annual to semi-annual compounding is larger than the increase from monthly to daily compounding. This might lead us to believe that although increasing the frequency of compounding will increase our result, there is an upper limit to this process.

To see this, let us examine the value of $\$1$ invested at 100% interest for 1 year.

These values do indeed appear to be approaching an upper limit. This value ends up being so important that it gets represented by its own letter, much like how $\pi$ represents a number.

Euler's Number: $e$

$e$ is the letter used to represent the value that $\begin{pmatrix}1+\dfrac1k\end{pmatrix}^k$ approaches as $k$ gets big. $e\approx2.718282$

Because $e$ is often used as the base of an exponential, most scientific and graphing calculators have a button that can calculate powers of $e$, usually labeled $e^x$. Some computer software instead defines a function $\mathrm{exp}(x)$, where $\mathrm{exp}(x)=e^x$.

Because $e$ arises when the time between compounds becomes very small, $e$ allows us to define continuous growth and allows us to define a new toolkit function. $f(x)=e^x$.

Continuous Growth Formula

Continuous growth can be calculated using the formula

$f(x)=ae^{rx}$ Where

$a$ is the starting amount

$r$ is the continuous growth rate

This type of equation is commonly used when describing quantities that change more or less continuously, like chemical reactions, growth of large populations, and radioactive decay.

For an example, continue to the next page.

Radon-222 decays at a continuous rate of 17.3% per day. How much will 100mg of Radon-222 decay to in 3 days?

Since we are given a continuous decay rate, we use the continuous growth formula. Since the substance is decaying, we know the growth rate will be negative: $r=-0.173$ $f(3)=100e^{-0.173(3)}\approx59.512$ mg of Radon-222 will remain.

Continuous growth is also often applied to compound interest, allowing us to talk about continuous compounding.

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If $\$1000$ is invested in an account earning 10% compounded continuously, find the value after 1 year.

Here, the continuous growth rate is 10%, so $r=0.10$. We start with $\$1000$, so $a=1000$. To find the value after 1 year, $f(1)=1000e^{0.10(1)}\approx\$1105.17$

Notice this is a $\$105.17$ increase for the year. As a percent increase, this is $\dfrac{105.17}{1000}=0.10517=10.517$ increase over the original $\$1000$.

Notice that this value is slightly larger than the amount generated by daily compounding in the table computed earlier.

The continuous growth rate is like the nominal growth rate (or APR) – it reflects the growth rate before compounding takes effect. This is different than the annual growth rate used in the formula $f(x)=a(1+r)^x$, which is like the annual percentage yield -- it reflects the actual amount the output grows in a year.

While the continuous growth rate in the example above was 10%, the actual annual yield was 10.517%. This means we could write two different looking but equivalent formulas for this account’s growth:

$f(t)=1000e^{0.10t}\;\;\;\;\;$using the 10% continuous growth rate

$f(t)=1000(1.10517)^t\;\;\;\;\;$using the 10.517% actual annual yield rate.

Graphs of Exponential Functions

Like with linear functions, the graph of an exponential function is determined by the values for the parameters in the function's formula.

To get a sense for the behavior of exponentials, let us begin by looking more closely at the function $f(x)=2^x$. Listing a table of values for this function:

Notice that:

1. This function is positive for all values of x.
2. As x increases, the function grows faster and faster (the rate of change increases).
3. As x decreases, the function values grow smaller, approaching zero.
4. This is an example of exponential growth.

Looking at the function $g(x)=\begin{pmatrix}\dfrac12\end{pmatrix}^x$

Note this function is also positive for all values of x, but in this case grows as x decreases, and decreases towards zero as x increases. This is an example of exponential decay. You may notice from the table that this function appears to be the horizontal reflection of $f(x)=2^x$ table. This is in fact the case:

$f(-x)=2^{-x}=(2^{-1})^x=\begin{pmatrix}\dfrac12\end{pmatrix}^x=g(x)$

Looking at the graphs also confirms this relationship:

Consider a function for the form $f(x)=ab^x$. Since a, which we called the initial value in the last section, is the function value at an input of zero, a will give us the vertical intercept of the graph. From the graphs above, we can see that an exponential graph will have a horizontal asymptote on one side of the graph, and can either increase or decrease, depending upon the growth factor. This horizontal asymptote will also help us determine the long run behavior and is easy to determine from the graph.

The graph will grow when the growth rate is positive, which will make the growth factor b larger than one. When it’s negative, the growth factor will be less than one.

Graphical Features of Exponential Functions

Graphically in the function $f(x)=ab^x$

1. $a$ is the vertical intercept of the graph
2. $b$ determines the rate at which the graph grows. When $a$ is positive, the function will increase if $b>1$ and the function will decrease if $0<b<1$
3. The graph will have a horizontal asymptote at $y=0$
4. The graph will be concave up if $a>0$; concave down if $a<0$.

The domain of the function is all real numbers

The range of the function is $(0,\infty)$

When sketching the graph of an exponential function, it can be helpful to remember that the graph will pass through the points $(0,a)$ and $(1,ab)$.

The value $b$ will determine the function's long run behavior:

If $b>1$, as $x\;\rightarrow\;\infty$, $f(x)\;\rightarrow\;\infty$ and as $x\;\rightarrow\;-\infty$, $f(x)\;\rightarrow\;0$.

If $0<b<1$, as $x\;\rightarrow\;\infty$, $f(x)\;\rightarrow\;0$ and as $x\;\rightarrow\;-\infty$, $f(x)\;\rightarrow\;\infty$.

If you want an example, continue to the next page.

Examples

Sketch a graph of $f(x)=4\begin{pmatrix}\dfrac13\end{pmatrix}^x$

This graph will have a vertical intercept at $(0,4)$, and pass through the point $\begin{pmatrix}1,\dfrac43\end{pmatrix}$. Since $b<1$, the graph will be decreasing towards zero. Since $a>0$, the graph will be concave up.

We can also see from the graph the long run behavior: as $x\;\rightarrow\;\infty$, $f(x)\;\rightarrow\;0$ and as $x\;\rightarrow\;-\infty$, $f(x)\;\rightarrow\;\infty$.

To get a better feeling for the effect of $a$ and $b$ on the graph, examine the sets of graphs below. The first set shows various graphs, where $a$ remains the same and we only change the value for $b$.

Notice that the closer the value of $b$ is to 1, the less steep the graph will be. In the next set of graphs, $a$ is altered and our value for $b$ remains the same.

Notice that changing the value for $a$ changes the vertical intercept. Since $a$ is multiplying the $b^x$ term, $a$ acts as a vertical stretch factor, not as a shift. Notice also that the long run behavior for all of these functions is the same because the growth factor did not change and none of these $a$ values introduced a vertical flip.

If you want to see one more example, continue to the last page.

Match each equation with its graph. below

1. $f(x)=2(1.3)^x$
2. $g(x)=2(1.8)^x$
3. $h(x)=4(1.3)^x$
4. $k(x)=4(0.7)^x$

The graph of $k(x)$ is the easiest to identify, since it is the only equation with a growth factor less than one, which will produce a decreasing graph. The graph of $h(x)$ can be identified as the only growing exponential function with a vertical intercept at $(0,4)$. The graphs of $f(x)$ and $g(x)$ both have a vertical intercept at $(0,2)$, but since $g(x)$ has a larger growth factor, we can identify it as the graph increasing faster.

Transformations of Exponential Graphs

While exponential functions can be transformed following the same rules as any function, there are a few interesting features of transformations that can be identified. The first was seen at the beginning of the section -- that a horizontal reflection is equivalent to a change in the growth factor. Likewise, since $a$ is itself a stretch factor, a vertical stretch of an exponential corresponds with a change in the initial value of the function.

Next consider the effect of a horizontal shift on an exponential function corresponds with a change in initial value of the function.

Lastly, consider the effect of a vertical shift on an exponential function. Shifting $f(x)=3(2)^x$ down 4 units would give the equation $f(x)=3(2)^x-4$, yielding the graph:

Notice that this graph is substantially different than the basic exponential graph. Unlike a basic exponential, this graph does not have a horizontal asymptote at $y=0$; due to the vertical shift, the horizontal asymptote has also shifted to $y=-4$. We can see that as $x\;\rightarrow\;\infty$, $f(x)\;\rightarrow\;\infty$ and as $x\;\rightarrow\;-\infty$, $f(x)\;\rightarrow\;-4$.

We have determined that a vertical shift is the only transformation of an exponential function that changes the graph in a way that cannot be achieved by altering the parameters $a$ and $b$ in the basic exponential function $f(x)=ab^x$.

Transformations of Exponentials

Any transformed exponential can be written in the form $f(x)=ab^x+c$ where $y=c$ is the horizontal asymptote.

Note that, due to the shift, the vertical intercept is shifted to $(0, a+c)$.

If you want some examples, continue to the next page.

Examples

Sketch a graph of $f(x)=-3\begin{pmatrix}\dfrac12\end{pmatrix}^x+4$

Notice that in this exponential function, the negative in the stretch factor -3 will cause a vertical reflection, and the vertical shift up 4 will move the horizontal asymptote to $y=4$. Sketching this as a transformation of $g(x)=\begin{pmatrix}\dfrac12\end{pmatrix}^x$.

Notice that while the domain of this function is unchanged, due to the reflection and shift, the range of this function is $(-\infty,4)$. As $x\rightarrow\infty$, $f(x)\rightarrow4$ and as $x\rightarrow-\infty$, $f(x)\rightarrow-\infty$.

Functions leading to graphs like the one above are common as models for learning and models of growth approaching a limit.

Find an equation for the graph sketched below.

Looking at this graph, it appears to have a horizontal asymptote at $y=5$, suggesting an equation of the form $f(x)=ab^x+5$. To find values for $a$ and $b$, we can identify two other points on the graph. It appears the graph passes through (0,2) and (-1,3), so we can use those points. Substituting in (0,2) allows us to solve for $a$

$2=ab^0+5$

$2=a+5$

$a=-3$

Substituting in (-1, 3) allows us to solve for $b$

$3=-3b^{-1}+5$

$-2=\dfrac{-3}b$

$-2b=-3$

$b=\dfrac32=1.5$

The final formula for our function is $f(x)=-3(1.5)^x+5$.