In this lesson, we will learn about the meaning of exponential functions, rules, and graphs.

Jonathan was reading a news article on the latest research made on bacterial growth.

He learned that an experiment was conducted with one bacterium. After the first hour, the bacterium doubled itself and was two in number. After the second hour, the number was four. At every hour the number of bacteria was increasing.

Before the first hour - 1 bacterium

After the first hour - 2 bacteria

After the second hour - 4 bacteria

After the third hour - 8 bacteria

After the fourth hour - 16 bacteria

Jonathan wanted to relate this to a math concept that he has learned in the recent past. He then flipped through his book and found that exponential functions could give him a clear picture of how the bacteria grew.

Try out the exponential function calculator given below with values of your choice! Enter the value of base in 'x' field and the power in 'n' field. 'n' can be positive or negative.

**Lesson Plan**

**What is an Exponential Function?**

The exponential function in math is of the following form.

\(\begin{align} f(x)\:=\:a^x\end{align}\) |

Where \(\begin{align} f(x)\:\end{align}\) is function of 'x'.

'a' is any number, except 0. It is called the base.

'x' is the exponent. It can take values starting from 0. The exponent can also be called the 'index'.

Exponential functions are very useful in real-life applications like finding the increase or decrease in population, rise in bacterial growth, compound interest in finance.

**What are the Rules of Exponential Functions?**

**Rule 1:**

For the function \(\begin{align} f(x)\:=\:a^x\end{align}\), when a = 1, the graph makes a horizontal line.

**Rule 2:**

A number can be raised to any power, with the limits starting from \(\begin{align} -\infty \:to\: \infty\end{align}\).

**Rule 3:**

In a given expression, the term with an exponent is to be evaluated first before performing the other operations like addition, subtraction.

For example, in the expression,

\(\begin{align} x\:=\: 2^3 + 3^2\end{align}\)

\(\begin{align} 2^3 , 3^2\end{align}\) are to be evaluated first and then the results are added.

**Rule 4:**

For the function \(\begin{align} f(x)\:=\:a^x\end{align}\), 'a' or the base can not be a negative number.

For example,

For the function \(\begin{align} f(x)\:=\:-3^5\end{align}\) is not valid.

**Rule 5:**

A number with a negative exponent is the reciprocal of the positive exponent of the number.

For example,

For the function \(\begin{align} f(x)\:=\:5^-2\end{align}\)

We can write ,

\(\begin{align}\:5^{-2}\ \:=\: \dfrac{1}{5^2}\end{align}\)

**Rule 6:**

When we multiply two terms with the same base, we can add their exponents.

For example,

In the expression, \(\begin{align}a^3 \times a ^5\end{align}\),

the bases are the same, so we add the exponents, to get the result as \(\begin{align}a^8\end{align}\)

**Rule 7:**

When we divide two terms with the same base, we can subtract their exponents.

For example, in the expression,

\(\begin{align}\dfrac{a^8}{a ^5}\end{align}\),

the bases are the same, so we subtract the exponents, to get the result as \(\begin{align}a^3\end{align}\).

**Rule 8:**

When there is more than one term inside a bracket, which is raised to an exponent, the power applies to each individual term.

For example, in the following expression,

\(\begin{align}(7a^2b^3c^4)^2\end{align}\), exponent \(\begin{align}2\end{align}\) applies to \(\begin{align}7,a^2,b^3,c^4\end{align}\) separately.

**Rule 9:**

As 'x' value increases in the fucntion f(x) starts moving to infinity. As 'x' value decreases, f(x) starts moving towards 0.

**Graphical Representation of Exponential Functions**

As we learnt earlier, in the exponential function f(x) = a^x, 'a' is the base and 'x' is the exponent.

There are three cases to be considered.

**Case 1: **Consider a = 1, \(\begin{align}f(x)\:=\:1^x\end{align}\). Taking different values and plotting the points, we obtain a straight line exactly at \(\begin{align}y\:=\:1\end{align}\).

**Case 2: **Consider a > 1, \(\begin{align}f(x)\:=\:2^x\end{align}\). Taking different values and plotting the points, we obtain a graph that is moving towards infinity.

**Case 3: **Consider a to be between \(\begin{align}0\end{align}\) and \(\begin{align}0\end{align}\), that is \(\begin{align}0 < a < 1\end{align}\).

Taking different values and plotting the points, we obtain a graph that crosees the y axis exactly at (0,1).

The graph line that is moving towards 0.

**What is Exponential Growth?**

Exponential growth defines the increase in growth of a quantity over a period of time, be it an hour or an year. If the quantity grows, then we call in exponential growth, if the quantity decreases then we call it exponential decay.

The formula to calculate exponential growth is as follows.

\(\begin{align}P\:=\:P_0e^{kt}\end{align}\) |

where,

\(\begin{align}P_0\:=\: Initial\: value\: of\: the\: quantity\end{align}\)

\(\begin{align}P\:=\:value\: after\: a\: certain\: time\: period\end{align}\)

\(\begin{align}k\:=\:exponential\: growth\: rate\: or\: decay\: rate\end{align}\)

\(\begin{align}t\:=time\:period\end{align}\)

\(\begin{align}e\:=Euler's\: number\end{align}\) ['e' is an exponential constant]

For example,

If \(\begin{align}P = 80\end{align}\), \(\begin{align}P_0 = 20\end{align}\).\(\begin{align}t = 5\end{align}\). Calculate the value of the exponential growth rate.

Using the formula,

\(\begin{align}P\:=\:P_0e^{kt}\end{align}\) and substituting the values, we get.

\(\begin{align}80 = 20 \times e^{k\times5}\end{align}\)

\(\begin{align}\dfrac{80}{20}\: = \:e^{5k}\end{align}\)

\(\begin{align}4\: = \:e^{5k}\end{align}\)

\(\begin{align}ln\:4\: = \:e^{5k}\end{align}\)

\(\begin{align}ln\:4\: = \:ln\:e^{5k}\end{align}\)

\(\begin{align}ln\:4\: = \:5k\end{align}\)

\(\begin{align}1.386\: = \:5k\end{align}\)

\(\begin{align}k\: = \:\dfrac{1.386}{5}\end{align}\)

\(\begin{align}k\: = \:0.2772\end{align}\)

- If the value of base in an exponential function is greater than 1, then the function increases exponentially.
- If the value of base in an exponential function is less than 1 but greater than 1, then the function decreases exponentially.
- If the value of base is 1, then the graph makes a line at y = 1 or y = a.
- The exponent power can be positive or negative.
- If the rate of increase in a quantity is positive then it is called exponential growth. If the rate of increase is negative then it is called exponential decay.

**Solved Examples**

Example 1 |

Simplify the following exponential functions.

\(\begin{align}f(x)\:=\:4^2.4^3\end{align}\)

\(\begin{align}f(x)\:=\:\dfrac{5^7}{5^3}\end{align}\)

**Solution**

In the function \(\begin{align}f(x)\:=\:4^2.4^3\end{align}\), the bases are the same which is \(\begin{align}4\end{align}\)

Since the operation is multiplication we add the powers of the bases.

On adding we get the expression simplified to,

\(\begin{align}f(x)\:=\:4^{2+3}\end{align}\)

\(\begin{align}f(x)\:=\:4^5\end{align}\)

In the function \(\begin{align}f(x)\:=\:\dfrac{5^7}{5^3}\end{align}\), the bases are the same which is \(\begin{align}5\end{align}\)

Since the operation is division we subtract the powers of the bases.

On subtractig we get the expression simplified to.

\(\begin{align}f(x)\:=\:5^{7-3}\end{align}\)

\(\begin{align}f(x)\:=\:5^4\end{align}\)

\(\begin{align}f(x)\:=\:4^2.4^3\end{align}\) \(\begin{align}f(x)\:=\:4^5\end{align}\) \(\begin{align}f(x)\:=\:\dfrac{5^7}{5^3}\end{align}\) \(\begin{align}f(x)\:=\:5^4\end{align}\) |

Example 2 |

Find the values of the following exponential functions.

\(\begin{align}f(x)\:=\:2^{-6}\end{align}\)

\(\begin{align}f(x)\:=\:{(3a^2b^3c^4})^3\end{align}\)

**Solution**

In the function \(\begin{align}f(x)\:=\:2^{-6}\end{align}\),

\(\begin{align}f(x)\end{align}\), can be written as,

\(\begin{align}f(x)\:=\:\dfrac{1}{2^6}\end{align}\), which simplifies to,

\(\begin{align}f(x)\:=\:\dfrac{1}{2\times2\times2\times2\times2\times2}\end{align}\)

\(\begin{align}f(x)\:=\:\dfrac{1}{64}\end{align}\)

In the exponential function, \(\begin{align}f(x)\:=\:{(3a^2b^3c^4})^3\end{align}\), power \(\begin{align}3\end{align}\) applies to all the terms inside the brackets.

So we multiply the power outside the brackets to all the powers of terms inside the brackets.

On simplification we get, \(\begin{align}f(x)\:=\:{3^3a^6b^9c^{12}}\end{align}\)

\(\begin{align}f(x)\:=\:2^{-6}\end{align}\)\(\begin{align}\:=\:\dfrac{1}{64}\end{align}\) \(\begin{align}f(x)\:=\:{(3a^2b^3c^4})^3\end{align}\) \(\begin{align}\:=\:{3^3a^6b^9c^{12}}\end{align}\) |

Example 3 |

The exponential growth equation of a certain type of bacteria is as follows. \(\begin{align}25000 = 10000\times e^{0.012t}\end{align}\). Find the number of days in which it grew exponentially.

**Solution**

We know that the formula for exponential growth is

\(\begin{align}P\:=\:P_0e^{kt}\end{align}\).

Comparing this from the given equation, we get,

\(\begin{align}P\: =\: 25000,\: P_0\: =\: 10000,\: k\: =\: 0.012 \end{align}\)

On substituting we get,

\(\begin{align}25000 = 10000 \times e^{0.012 \times t }\end{align}\)

\(\begin{align}\dfrac{25000}{10000} = e^{0.012 \times t }\end{align}\)

\(\begin{align}2.5= e^{0.012 \times t }\end{align}\)

\(\begin{align}ln\:2.5 = ln e^{0.012 \times t }\end{align}\)

\(\begin{align}0.9162 = 0.012 \times t \end{align}\)

\(\begin{align}0.9162 = 0.012 \times t \end{align}\)

\(\begin{align}\dfrac {0.9162}{0.012 } \:=\: t \end{align}\)

\(\begin{align}t\:=\: 76.3 \end{align}\)

Approximately, \(\begin{align}76\end{align}\) days.

Time period of exponential growth \(\begin{align}\:=\:76\end{align}\) days. |

Example 4 |

Find the exponential growth in money if a sum of \(\begin{align}$2000\end{align}\) was invested for a period of \(\begin{align}15\end{align}\) years at a rate of \(\begin{align}9\%\end{align}\). Assume that the interest in compounded monthly.

**Solution**

Given,

P = $2000

t = 15 years

n = 12

r = 9%

The fomula to calculate compound interest is as follows.

\(\begin{align}A\:=\:2000(1 + \dfrac{0.09}{12})^{12\times15}\end{align}\)

\(\begin{align}A\:=\:2000(1 + 1.0075^{180}\end{align}\)

\(\begin{align}A\:=\:2000(1 + 3.838)\end{align}\)

\(\begin{align}A\:=\:2000(4.838)\end{align}\)

\(\begin{align}A\:=\:9676\end{align}\)

\(\begin{align}Amount\: got\: after\: 15\: years\: \:=\:9676\end{align}\) |

**Interactive Questions**

**Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.**

- A bacterial growth of a certain type of bacteria started at 80 at 10:00 AM. It grew exponentially to 400 at 5:00 PM. Design an exponential growth equation based on this information.
- Find the value of 'x' in 75 = 15 e
^{0.75x}.

**Let's Summarize**

The lesson targeted on the fascinating concept of exponential functions. By now you can find the solve exponents with positive and negative powers, perform arithmetic operations on exponential functions with the same base, read and plot the graph for an exponential function. Hope you enjoyed reading the lesson!

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**Frequently Asked Questions(FAQs)**

## 1) What is the relationship between logarithmic functions and exponential functions?

Logarithmic functions are the inverse of exponential functions. If an exponential function is of the form y = b^x, then, the logarithmic function will be x = b^y.

## 2) How are exponential equations used in real life?

Exponential growth can be used to track the growth of any infection, bacteria growth and decay, increase in population, in finance we use it with compound interest.

## 3) Why are exponential functions important?

Exponential functions are important to find out the growth and decay of a quantity. For example, we can find how the population of a state has grown exponentially.