Exponential Function Formula

Algebra is a large field of mathematics. Algebra, in a nutshell, is the study of mathematical symbols and the rules for manipulating these symbols in formulas, it is a common thread that runs through practically all of mathematics.

Exponential Function

An exponential function is a mathematical function of the shape f (x) = a^x, where ‘x’ is a variable and ‘a’ is a consistent this is the function’s base and needs to be more than 0. The transcendental wide variety e, that’s about the same as 2.71828, is the most customarily used exponential function basis.

For Examples,

• f(x) = 2^x
• f(x) = (1/2)^x
• f(x) = 3e^2x
• f(x) = 4 (3)^-0.5x

Formula for Exponential Growth:

At first, the quantity grows very slowly, then rapidly in exponential growth. Over time, the rate of change accelerates. With the passage of time, the rate of growth accelerates. The quick expansion is described as an “exponential increase.” 

The exponential growth formula is as follows:

y = a ( 1+ r )^x

Formula for Exponential Decay

In Exponential Decay, the quantity decreases rapidly at first, then gradually. The rate of change slows over time. The rate of change slows with the passage of time. The rapid rise was supposed to create a “exponential decline.” 

The formula for exponential growth is as follows:

y = a ( 1- r )^x

Exponential Series

• The following power series can be used to define the real exponential function.

e^x = ∑^∞_n=0 xⁿ/n! = (1/1) + (x/1) + (x²/2) + (x³/6) + …

• Some other exponential functions’ expansions are illustrated below,

e = ∑^∞_n=0 xⁿ/n! = (1/1) + (1/1) + (1/2) + (1/6) + …

e^-1 = ∑^∞_n=0 xⁿ/n! = (1/1) – (1/1) + (1/2) – (1/6) + …

Exponential Function Rules

The following are some important exponential rules:

For any real numbers x and y, if a>0 and b>0, the following is true:

• a^x a^y = a^x+y
• a^x/a^y = a^x-y
• (a^x)^y = a^xy
• a^xb^x=(ab)^x
• (a/b)^x= a^x/b^x
• a⁰=1
• a^-x= 1/ a^x

Equality Property of Exponential Function

If two exponential functions with the same bases are equal, their exponents are likewise equal, according to the equality property of exponential functions. i.e.,

b^x₁= b^x₂ ⇔ x₁= x₂

Exponential Function Derivative

The differentiation formulas that are used to obtain the exponential function’s derivative,

d/dx (e^x) = e^x

d/dx (a^x) = a^x · ln a

Integration of Exponential Function

The integral of an exponential function is calculated using integration formulae.

∫ e^x dx = e^x + C

∫ a^x dx = a^x / (ln a) + C

Sample Problems

Problem 1: In 2010, a town had 100,000 inhabitants. How many citizens will there be in ten years if the population grows at an annual rate of 8%? 

Solution:

The initial population is, a = 100,000.

The rate of growth is, r = 8% = 0.08.

The time is, x = 10 years.

Using the exponential growth formula,

f(x) = a (1 + r)^x

f(x) = 100000(1 + 0.08)¹⁰

f(x) ≈ 215,892

Problem 2: Carbon-14 has a half-life of 5,730 years. What is the quantity of carbon left after 2000 years if there were 1000 kilos of carbon at the start?

Solution:

Using the given data, we can say that carbon-14 is decaying and hence we use the formula of exponential decay.

P = P₀ e^{-k t} … (1),

Here, P₀= initial amount of carbon = 1000 grams.

It is given that the half-life of carbon-14 is 5,730 years. It means

P = P₀/ 2 = 1000 / 2 = 500 grams.

Substitute all these values in (1),

500 = 1000 e^{-k (5730)}

Dividing both sides by 1000,

0.5 = e^{-k (5730)}

Taking “ln” on both sides,

ln 0.5 = -5730k

Dividing both sides by -5730,

k = ln 0.5 / (-5730) ≈ 0.00012097

We have to find the amount of carbon that is left after 2000 years. Substitute t = 2000 in (1),

P = 1000 e ^{-(0.00012097) (2000)} ≈ 785 grams.

Problem 3: Simplify the following exponential expression: 3^x – 3^x+2.

Solution:

Given exponential equation: 3^x – 3^x+2

By using the property: a^x a^y = a^x+y

Hence, 3^x+2 can be written as 3^x.3²

Thus the given equation is written as:

3^x – 3^x+2 = 3^x – 3^x·9

Now, factor out the term 3^x

3^x – 3^x+2 = 3^x – 3^x·9 = 3^x(1 – 9)

3^x – 3^x+2 = 3^x(-8)

3^x – 3^x+2 = -8(3^x)

Therefore, the simplification of the given exponential equation 3^x-3^x+1 is -8(3^x).

Problem 4: Solve the exponential equation: (¼)^x = 64.

Solution:

Given exponential equation is:  

(¼)^x = 64

Using the exponential rule (a/b)^x = a^x/b^x , we get;

1^x/4^x = 43

1/4^x = 43 [since 1^x = 1]

(1)(4^-x) = 4³

4^-x = 4³

Here, bases are equal.

So, 

x = -3

Problem 5: Simplify the exponential function 2^x – 2^x+1.

Solution:

Given exponential function: 2^x – 2^x+1

By using the property: a^x a^y = a^x+y

Hence, 2^x+1 can be written as 2^x. 2

Thus the given function is written as:

2^x – 2^x+1 = 2^x – 2^x. 2

Now, factor out the term 2^x

2^x – 2^x+1 = 2^x – 2^x. 2 = 2^x(1-2)

2^x – 2^x+1 = 2^x(-1)

2^x – 2^x+1 = – 2^x

Therefore, the simplification of the given exponential function 2^x – 2^x+1 is  – 2^x.