📊 Exponential Function and Its Applications

Sahabat Smaridasa — In this article, we will discuss the topic of Exponential Function and Its Applications. An exponential function is a function that contains an exponential form, meaning it contains a power where the exponent consists of variables. As for the application of exponential functions, one of them relates to "growth" and "decay," which you can learn in the Grade XII high school mathematics curriculum.

To make it easier to study Exponential Functions and Their Applications, we must first master the properties of exponents. In this discussion, we will first cover the exponential function, then continue with the application of exponential functions. Let's go through the material below.

📐 Exponential Function

Definition

Below are the forms of exponential functions:

Simple exponential function: \[ f(x) = a^x \] where $ a $ is the base and $ x $ is the exponent.

Complex exponential function: \[ f(x) = b \cdot a^{g(x)} + c \] where $ a $ is the base and $ g(x) $ is the exponent.

Example 1

Below are some examples of exponential functions:
a) $ f(x) = 2^x $    b) $ g(x) = 3^{5x} $    c) $ h(x) = \left( \frac{1}{5} \right)^x $
d) $ f(x) = 3 \cdot 5^x $    e) $ f(x) = 2 \cdot 3^x + 5 $    f) $ f(x) = 3^{x^2+2x-8} $
g) $ f(x) = 2 \cdot 5^{x^3 - x + 1} - 1 $

Note: All of the above are examples of exponential functions because they contain variables in the exponent.

Example 2

Given the exponential function $ f(x) = 3^{x+1} - 2 $. Determine the value of $ f(1) $.

Solution:

Determine the value of $ f(1) $ by substituting $ x = 1 $: \[ \begin{aligned} x = 1 \rightarrow f(x) &= 3^{x+1} - 2 \\ f(1) &= 3^{1+1} - 2 \\ &= 3^2 - 2 \\ &= 9 - 2 \\ &= 7 \end{aligned} \] Answer: $ f(1) = 7 $. $\heartsuit$

Example 3

Given an exponential function $ f(x) = 2^{x-1} - 1 $. If $ f(a) = 31 $, then what is the value of $ a^2 - 30 $?

Solution:

From the function $ f(x) = 2^{x-1} - 1 $, we have: \[ f(a) = 2^{a-1} - 1 \] Determine the value of $ a $ from $ f(a) = 31 $: \[ \begin{aligned} f(a) &= 31 \\ 2^{a-1} - 1 &= 31 \\ 2^{a-1} &= 32 \\ 2^{a-1} &= 2^5 \quad \text{(cancel the bases)} \\ a - 1 &= 5 \\ a &= 6 \end{aligned} \] So: \[ a^2 - 30 = 6^2 - 30 = 36 - 30 = 6 \] Answer: $ a^2 - 30 = 6 $. $\heartsuit$

Example 4

Given an exponential function $ f(x) = 3^{2x} $. Express $ f(3a+b-c) $ in terms of $ f(a) $, $ f(b) $, and $ f(c) $.

Solution:

Exponent properties: $ a^{m+n} = a^m \cdot a^n $ and $ a^{m-n} = \frac{a^m}{a^n} $.

From the initial function $ f(x) = 3^{2x} $, we get: \[ f(a) = 3^{2a}, \quad f(b) = 3^{2b}, \quad f(c) = 3^{2c} \] To express $ f(3a+b-c) $ in terms of $ f(a) $, $ f(b) $, and $ f(c) $, we need to convert it to the forms above.

Modify and solve: \[ \begin{aligned} f(x) &= 3^{2x} \\ f(3a+b-c) &= 3^{2(3a+b-c)} \\ &= 3^{6a+2b-2c} \\ &= \frac{3^{6a} \cdot 3^{2b}}{3^{2c}} \\ &= \frac{(3^{2a})^3 \cdot 3^{2b}}{3^{2c}} \\ &= \frac{(f(a))^3 \cdot f(b)}{f(c)} \end{aligned} \] Answer: \[ f(3a+b-c) = \frac{(f(a))^3 \cdot f(b)}{f(c)} \] $\heartsuit$

📈 Applications of Exponential Functions

Growth and Decay Models

One application of exponential functions is in growth and decay models. You can read the complete material in the articles "growth in mathematics" and "decay in mathematics". However, for certain problems, the exponential function form is usually given in advance. The general form of the exponential function for growth and decay is: \[ A_t = A_0 \cdot (r)^t \]
Where:
$ A_t = $ the amount of growth or decay at time $ t $
$ A_0 = $ the initial amount of growth or decay at the beginning of the period
$ r = $ the ratio (rate of change)

Example 5

In biology, there is the concept of growth of a certain type of amoeba. Suppose its growth follows the exponential function $ A_t = A_0 \cdot (2)^t $, where $ A_0 $ is the initial number of amoebas at the start of observation and $ t $ is the observation time (in minutes). If at the beginning of observation at 09:00 there are 100 amoebas, determine the number of amoebas after observation at 09:10.

Solution:

Given: $ A_0 = 100 $ amoebas.
From 09:00 to 09:10, $ t = 10 $ minutes.

Determine the number of amoebas at $ t = 10 $: \[ \begin{aligned} A_t &= A_0 \cdot (2)^t \\ A_{10} &= 100 \cdot (2)^{10} \\ &= 100 \cdot 1024 \\ &= 102,400 \end{aligned} \] Answer: There will be 102,400 amoebas at the observation at 09:10. $\heartsuit$

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Thus concludes our discussion on Exponential Functions and Their Applications along with examples. Next, please also read other related materials such as the graph of exponential and logarithmic functions.

💡 Tip: The best way to master exponential functions and their applications is through continuous practice. Try various problems involving different bases and exponents, as well as real-world applications like growth and decay.