Exponential Function

Exponential functions are mathematical functions that increase or decrease rapidly. The rate at which they grow or shrink depends on the value of the function itself, making these perfect for describing things that grow or shrink rapidly, such as populations, money investments, etc.

General form of exponential functions is

f(x) = a ⋅ b^x

• a is a constant called the coefficient.
• b is the base , which must be a positive real number other than 1.
• x is the exponent, which is typically a variable.

Key features:

• If b > 1, it increases rapidly as x increases.
• If 0 < b < 1, it decreases rapidly as x increases.
• The graph of an exponential function never touches the x-axis.

Exponential Growth

In Exponential Growth, a quantity progresses rapidly. An exponentially growing function has an increasing graph. It 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

• r is the growth rate.

Exponential Decay

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 rate

Exponential Rules

Exponential Function Derivative

• For f(x) = eˣ, its derivative is, d/dx (eˣ) = eˣ.
• For f(x) = ax, its derivative is, d/dx (a^x) = a^x · ln a

Exponential Function Integration

• For f(x) = eˣ, its integration is ∫eˣ dx = eˣ + C
• For f(x) = ax, its integration is, ∫a^x dx = a^x / (ln a) + C

Solved Examples

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

Solution:

Given exponential function: 5^x - 5^x+3

From the properties of an exponential function, we have a^x × a^y = a^{(x + y)}

So, 5^x+3 = 5^x × 5³ = 125×5^x

Now, the given function can be written as

5^x - 5^x+3 = 5^x - 125 × 5^x
= 5^x(1 - 125)
=5^x(-124)
= -124(5^x)

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

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 a^x × a^y = 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 a^x × a^y = 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.

Practice Questions

Question 1: Calculate the value of f(x) for f(x) = 3.2^x when x = 4.

Question 2: Given the exponential function g(x) = 5(0.5)ˣ, sketch the graph of the function. Indicate the behavior of the function as x increases and as x decreases. Identify any asymptotes and intercepts.

Question 3: A population of bacteria doubles every hour. If the initial population is 200 bacteria, express the population P as an exponential function of time t in hours. Then, find the population after 6 hours.

Question 4: Solve for x in the exponential equation 10 × 3^x = 90.