Mastering Exponential Functions: A Step-by-Step Guide for Success 2024

Exponents are a fundamental concept in mathematics. In this blog post, we’ll cover the basics of exponents, exponential functions, delve into more advanced topics, and explore practical applications. Let’s start with the basic definition of exponents.

ⓘ This post was written after studying with lots of information like books, internet sites, etc. Except in some cases where it is specifically stated, images are made by R & Python using RStudio or VS code.

What is an Exponent?

An exponent refers to the number of times a number is multiplied by itself. For example, (2^3) means (2 *2 *2 = 8). Here, ‘2‘ is the base and ‘3‘ is the exponent.

$2^3 = 2 \times 2 \times 2 = 8$

Exponent Laws

Multiplication Law

When multiplying two exponents with the same base, you add the exponents:

$a^m \times a^n = a^{m+n}$

Examples:

$2^3 \times 2^4 = 2^{3+4} = 2^7 = 128$

$5^2 \times 5^3 = 5^{2+3} = 5^5 = 3125$

Division Law

When dividing two exponents with the same base, you subtract the exponents:

$\frac{a^m}{a^n} = a^{m-n}$

Examples:

$\frac{2^5}{2^3} = 2^{5-3} = 2^2 = 4$

$\frac{10^6}{10^2} = 10^{6-2} = 10^4 = 10000$

Power of a Power Law

When raising an exponent to another exponent, you multiply the exponents:

$(a^m)^n = a^{mn}$

Examples:

$(3^2)^3 = 3^{2 \times 3} = 3^6 = 729$

$(4^3)^2 = 4^{3 \times 2} = 4^6 = 4096$

Extended Exponents

Negative Exponents

Negative exponents represent the reciprocal of the base raised to the positive exponent:

$a^{-n} = \frac{1}{a^n}$

Examples:

$2^{-3} = \frac{1}{2^3} = \frac{1}{8}$

$5^{-2} = \frac{1}{5^2} = \frac{1}{25}$

Fractional Exponents

Fractional exponents represent roots. For example:

$a^{\frac{m}{n}} = \sqrt[n]{a^m}$

Examples:

$8^{\frac{1}{3}} = \sqrt[3]{8} = 2$

$16^{\frac{1}{4}} = \sqrt[4]{16} = 2$

Zero Exponents

Any non-zero base raised to the power of zero is 1:

$a^0 = 1$

Exponential Functions

Definition

An exponential function is defined as:

$f(x) = a^x$

Graph of an Exponential Function

Exponential functions grow rapidly. Here’s the graph of

$2^x$

$3^x$

This graphic was created using VS Code, and the source code is available at the bottom of this post.

Growth Rate and Behavior

$2^x$

The blue curve represents the function 2^x.
As x increases, the function value increases at a steady exponential rate.
When x is negative, 2^x approaches zero but remains positive.
Around x = 0, the function value is 1, as any number raised to the power of 0 equals 1.
As x moves into positive values, the function grows rapidly but at a slower rate compared to 3^x.

$3^x$

The orange curve represents the function $3^x$ .
Similar to 2^x, 3^x also grows exponentially but at a much faster rate.
For negative x, 3^x also approaches zero but remains positive.
At x = 0, the function value is 1, same as 2^x.
As x becomes positive, 3^x increases significantly faster than 2^x.

Key Differences

Growth Rate:
- The primary difference between these two curves lies in their base numbers. 3^x grows faster than 2^x because 3 is larger than 2.
- This means that for any given positive value of x, 3^x will always be greater than 2^x.
Intersection and Divergence:
- Both functions intersect at the point (0, 1), as any number raised to the power of 0 equals 1.
- Beyond this point, 3^x diverges away from 2^x more rapidly as x increases, illustrating its faster growth rate.

Applications of Exponents

Compound Interest Calculation

Compound interest can be calculated using the formula:

$A = P(1 + \frac{r}{n})^{nt}$

Explanation:

A: The amount of money accumulated after n years, including interest.
P: The principal amount (the initial amount of money).
r: The annual interest rate (in decimal form).
n: The number of times interest is compounded per year.
t: The number of years the money is invested or borrowed for.

Detailed Interpretation:

Principal (P): This is the starting amount of money before any interest is added.
Annual Interest Rate (r): This is the percentage of the principal that is paid as interest each year, converted to a decimal. For example, 5% becomes 0.05.
Number of Times Compounded (n): This tells us how frequently the interest is added to the principal in one year. For example, if interest is compounded quarterly, then n = 4.
Time (t): This is the total time the money is invested or borrowed for, measured in years.
Compound Interest Formula:

The formula calculates how much money (A) you will have after a certain number of years (t), given the principal (P), annual interest rate (r), and the number of times the interest is compounded per year (n).
The term $\left(1 + \frac{r}{n}\right)^{nt}$ represents the growth factor of the investment. It shows how much the money grows due to compounding interest over time.

Example:

If you invest $1000 at an annual interest rate of 5% (0.05) compounded quarterly (n=4) for 10 years (t=10), the calculation would be:

[ $A = 1000 \left(1 + \frac{0.05}{4}\right)^{4 \times 10}$ ]
[ $A = 1000 \left(1 + 0.0125\right)^{40}$ ]
[ $A = 1000 \left(1.0125\right)^{40}$ ]
[ $A \approx 1647.01$ ]

So, after 10 years, you would have approximately $1647.01.

Population Growth Model

Population growth can be modeled with an exponential function:

$P(t) = P_0 e^{rt}$

Explanation:

P(t): The population at time t.
$<strong>P_0</strong>$ : The initial population (at time t = 0).
e: The base of the natural logarithm (approximately equal to 2.71828).
r: The growth rate (in decimal form).
t: The time period over which the population grows.

Detailed Interpretation:

Initial Population ( $P_0$ ): This is the starting number of individuals in the population.
Growth Rate (r): This represents how quickly the population is growing, expressed as a decimal. For example, a growth rate of 3% is 0.03.
Time (t): This is the time period over which the population is measured, typically in years.
Exponential Growth Formula:

The formula calculates the population at a future time (P(t)), based on the initial population ( $P_0$ ), the growth rate (r), and the time period (t).
The term ( $e^{rt}$ ) represents the factor by which the population grows due to the exponential nature of growth. Exponential growth means the population increases by a certain percentage over equal time intervals, leading to increasingly larger increments.

Example:

If the initial population is 1000 ( $P_0$ ), the annual growth rate is 3% (0.03), and you want to find the population after 5 years (t=5), the calculation would be:

[ $P(t) = 1000 \times e^{0.03 \times 5}$ ]
[ $P(t) = 1000 \times e^{0.15}$ ]
[ $P(t) \approx 1000 \times 1.1618$ ]
[ $P(t) \approx 1161.83$ ]

So, after 5 years, the population would be approximately 1161.83 individuals.

Conclusion

In this post, we covered the basics of exponents, various exponent laws, extended exponent concepts, exponential functions, and practical applications. Exponents are a key mathematical concept with wide-ranging applications. Keep practicing and exploring!

Simple Common Log Calculator

Simple Common Log Calculator

# Python Code for Graphs

Here’s the Python code to generate the graphs:

import matplotlib.pyplot as plt
import numpy as np

x = np.linspace(-2, 2, 400)
y1 = 2 ** x
y2 = 3 ** x

plt.figure(figsize=(10, 6))

plt.plot(x, y1, label=r'$2^x$')
plt.plot(x, y2, label=r'$3^x$')

plt.axhline(0, color='black',linewidth=0.5)
plt.axvline(0, color='black',linewidth=0.5)
plt.grid(color = 'gray', linestyle = '--', linewidth = 0.5)

plt.title('Exponent Functions')
plt.xlabel('x')
plt.ylabel('y')
plt.legend()
plt.show()

Would you like to run the Python code and see the graph directly? If so, please click the button below to install VS Code first. It is explained in Korean, but you can right-click on the post body to translate it into English.

Installing ‘VS Code’ for Python source code execution.

Excercise of Python source code execution.

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Mastering Exponential Functions: A Step-by-Step Guide for Success 2024

What is an Exponent?