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Exponential Functions

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  • Last Updated : 12 Sep, 2022
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In mathematics, exponential functions and exponential formulas help in solving the calculations of large numbers and are used in different real-life situations. For example, using exponential functions, we can determine the population growth of a city, the rate of growth of bacteria in a culture, the half-life, the radioactive decay of the isotopes of radioactive elements, etc. As the name suggests, an exponential function is said to be a function that involves exponents. An exponential function is classified into two types, i.e., exponential growth and exponential decay. In this article, we will discuss the definition of an exponential function, its graph, types, and also exponential formulas along with some solved examples.

Exponential Function

As the name suggests, an exponential function is said to be a function that involves exponents. A mathematical function is in the form of f(x) = ax, where “a” is called the base of the function, which is a constant greater than 0, and “x” is the exponent of the function, which is a variable. When x > 1, the function f(x) increases with increasing x values. Mostly, a transcendental number denoted by e is used as the base of an exponential function. The value of “e” is approximately equal to 2.71828. The curve of an exponential function depends on the value of x. The domain of an exponential function is a set of all real numbers R, while its range is a set of all positive real numbers.

The formula for an exponential function is given as follows:

f(x) = ax, where a>0 and a ≠ 1 and x ∈ R

The exponential function is classified into two types based on the growth or decay of an exponential curve, i.e., exponential growth and exponential decay.

Exponential Growth

As the name suggests, in exponential growth, a quantity increases very slowly at first and then progresses rapidly. An exponentially growing function has an increasing graph. The exponential growth formula can be used to illustrate economic growth, population expansion, compound interest, growth of bacteria in a culture, population increases, etc.

The formula for exponential growth is:

y = a(1 + r)x

where,

r is the growth percentage

Exponential Decay

As the name suggests, in exponential decay, a quantity decreases very rapidly at first and then fades gradually. An exponentially decaying function has a decreasing graph. The concept of exponential decay can be applied to determine half-life, mean lifetime, population decay, radioactive decay, etc.

The formula for exponential decay is:

y = a(1 – r)x

where,

r is the decay percentage.

Exponential Function Graph

The image given below represents the graphs of the exponential functions y = ex and y = e-x. From the graphs, we can understand that the ex graph is increasing while the graph of e-x is decreasing. The domain of both functions is the set of all real numbers, while the range is the set of all positive real numbers.

Exponential Curve

Exponential Curve

For an exponential function y = ax (a>1), the logarithm of y to base e is x = logay, which is the logarithmic function. Now, observe the graph of the natural logarithmic function y = logex. From the graph, we can notice that a logarithmic function is only defined for positive real values. As the logarithmic function is not defined for negative values, its domain is the set of all positive real numbers.

Curve comparison of log x and e^x

Curve comparison of log x and ex

Exponential graph of f(x) = 2x

Let us consider an exponential function f(x) = 2x.

x

-3

-2

-1

0

1

2

3

f(x)= 2x f(-3) = 2-3 = 1/8 = 0.125 f(-2) = 2-2 = 1/4 = 0.25 f(-1) = 2-1 = 1/2 = 0.5 f(0) = 20 = 1 f(1) = 21 = 2 f(2) = 22 = 4 f(3)= 23 = 8
Graph of 2^x

Graph of 2x

From the graph, we can observe that the graph of f(x) = 2x is upward-sloping, increasing faster as the value of x increases. The graph formed is increasing and is also smooth and continuous. The graph lies above the X-axis and passes through (0, 1). As x approaches negative infinity, the graph becomes arbitrarily close to the X-axis. The domain of an exponential function holds all real values, whereas its range contains all values greater than zero (y>0).

Exponential graph of f(x) = 2-x

Let us consider an exponential function f(x) = 2-x = (1/2)x.

x

-3

-2

-1

0

1

2

3

f(x) = (1/2)x f(-3) = (1/2)-3=8 f(-2)=(1/2)-2=4 f(-1)=(1/2)-1=2 f(0)=(1/2)0=1 f(1)=(1/2)1= 0.5 f(2)=(1/2)2= 1/4=0.25 f(3)=(1/2)3=1/8= 0.125
Graph of 2^{-x}

Graph of 2-x

From the graph, we can observe that the graph of f(x) = 2-x is downward-sloping, decreasing faster as the value of x increases. The graph formed is decreasing and is also smooth and continuous. The graph lies above the X-axis and passes through (0, 1). As x approaches positive infinity, the graph becomes arbitrarily close to the X-axis. The domain of an exponential function holds all real values, whereas its range contains all values greater than zero (y>0).

Graph of 10^x

Graph of 10x

The image given above represents the graph of exponents of x. One can observe that as the exponent of the function increases, the curve is getting steeper. So, we can conclude that the nature of a polynomial function depends on its degree. Therefore, for all x > 1, a function y = fn(x) increases as the value of “n” increases. Thus, any polynomial function with a higher degree has a higher growth. However, a function f(x) = ax (where a > 1) grows faster than a polynomial function. Hence, for all positive integers n, the function f (x) grows faster than the function fn (x).

Exponential Formulae

The following are some exponential formulas for exponential functions.

Power of zero rule a0 =1
Negative power rule a-x = 1/ax
Product Rule ax × ay = a(x + y)
Quotient Rule ax/ay = a(x – y)
Power of power rule (ax)y = axy
Power of a product power rule ax × bx=(ab)x
Power of a fraction rule (a/b)x= ax/bx
Fractional exponent rule

(a)1/y = y√a

(a)x/y = y√(ax)

Solved Examples on Exponential Formula

Example 1: Simplify the exponential function 5x – 5x+3.

Solution:

Given exponential function: 5x – 5x+3

From the properties of an exponential function, we have ax × ay = a(x + y)

So, 5x+3 = 5x × 53 = 125×5x

Now, the given function can be written as

5x – 5x+3 = 5x – 125×5x

= 5x(1 – 125)

=5x(–124)

= –124(5x)

Hence, the simplified form of the given exponential function is –124(5x).

Example 2: Find the value of x in the given expression: 43× (4)x+5 = (4)2x+12.

Solution:

Given,

43× (4)x+5 = (4)2x+12

From the properties of an exponential function, we have ax × ay = a(x + y)

⇒ (4)3+x+5 = (4)2x+12

⇒(4)x+8 = (4)2x+12

Now, as the bases are equal, equate the powers.

⇒ x+8 = 2x+12

⇒ x – 2x = 12 – 8

⇒ – x = 4

⇒ x = –4

Hence, the value of x is –4.

Example 3: Simplify: (3/4)–6 × (3/4)8.

Solution:

Given: (3/4)–6 × (3/4)8

From the properties of an exponential function, we have ax × ay = a(x + y)

Thus, (3/4)–6×(3/4)8 = (3/4)(–6+8)

= (3/4)2

= 3/4 × 3/4 = 9/16

Hence, (3/4)–6 × (3/4)8 = 9/16.

Example 4: In the year 2009, the population of the town was 60,000. If the population is increasing every year by 7%, then what will be the population of the town after 5 years?

Solution:

Given data:

Population of the town in 2009 (a) = 60,000

Rate of increase (r) = 7%

Time span (x) = 5 years

Now, by the formula for the exponential growth, we get,

y = a(1+ r)x

= 60,000(1 + 0.07)5

= 60,000(1.07)5

= 84,153.1038 ≈ 84,153.

So, the population of the town after 5 years will be 84,153.

Example 5: Solve the exponential equation: (1/6)x–5 = 216.

Solution:

Given exponential equation is:

(1/6)x–5 = 216

⇒ (1/6)x–5 = 63

From exponent formulas, we have a–x = 1/ax

So, (1/6)x–5 = 6–(x–5)

6–(x–5) = 63

Now, as the bases are equal, equate the powers.

⇒ – (x – 5) = 3

⇒ –x + 5 = 3

⇒ –x = 3 – 5 = –2

⇒ x = 2

Hence, the value of x is 2.

FAQs on Exponential Formula

Question 1: Define an exponential function.

Answer:

An exponential function is defined as a mathematical function with the formula f(x) = ax, where “x” is a variable and is known as the exponent of the function, and “a” is a constant greater than zero and is known as the base of the function.

Question 2: Mention some properties of an exponential function.

Answer:

The domain of an exponential function holds all real values, whereas its range contains all values greater than zero (y>0). Its graph is upward-sloping, increasing faster as the value of x increases. The graph lies above the X-axis and passes through (0, 1). As x approaches negative infinity, the graph becomes arbitrarily close to the X-axis.

Question 3: Mention some exponential formulas.

Answer:

The following are some exponential formulas for exponential functions. They hold true if a > 0 and for all real values of m and n.

  • ax × ay = a(x + y)
  • ax/ay = a(x – y)
  • a0 = 1
  • a-x = 1/ax
  • (am)n = amn

Question 4: What are the different types of exponential functions?

Answer:

The exponential function is classified into two types based on the growth or decay of an exponential curve, i.e., exponential growth and exponential decay. As the name suggests, in exponential growth, a quantity increases very slowly at first and then progresses rapidly, while in exponential decay, a quantity decreases very rapidly at first and then fades gradually.

Question 5: What are the formulae for exponential growth and exponential decay?

Answer:

The formula for exponential growth is:

y = a(1+ r)x where r is the growth percentage.

The formula for exponential decay is:

y = a(1 – r)x, where r is the decay percentage.


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