Exponential Function (Practice Questions)

An exponential function is a mathematical function in which the variable is present in the exponent. It is generally written as f(x) = a^x where a > 0 and a ≠ 1.

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.