Algebra Question

This assignment serves as an opportunity to assess your ability to identify exponential and logarithmic functions. Among the various functions you’ve studied, exponential and logarithmic functions hold particular significance when it comes to representing practical situations marked by non-linear and rapid changes. Recognizing logarithms as the inverse of exponential functions will enable you to solve equations involving exponential functions using logarithmic properties.

Within this assignment, you will delve into the properties of logarithms and acquire the skill to express scenarios using either exponential or logarithmic functions.

You are required to complete all the 3 tasks in this assignment, answer the following questions, and show stepwise calculations. When you are instructed to make a graph in this assignment, please use GeoGebra graphing tool for drawing the graphs.

Task 1.

Please answer the following questions related to exponential and logarithmic functions:

(i)What are exponential and logarithmic functions? How are they related? What are their key factors (Explain the variables used in the definitions of these functions)? Discuss their domain and range.

(ii) What is the difference between exponential, logarithmic, and power functions? Provide one mathematical example for each and illustrate the differences of growth patterns and any special points (such as asymptotes, intercepts, and zeros), if applicable. Graph the examples. 

(iii)How to explain if a function has exponential growth?

(iv)Between exponential and logarithmic functions, which one grows faster? Provide an explanation for your answer.

(v) Write the observations of growth patterns and special points (if any) by drawing the graphs for the examples given 

Task 2. Before working on task 2, please read the following reading:?

Reading section?4.1- Exponential Growth and Decay?of the following textbook will help you in understanding the concepts better.

Yoshiwara, K. (2020).?Modeling, functions, and graphs.?American Institute of Mathematics.?https://yoshiwarabooks.org/mfg/frontmatter.html

Write the logarithmic properties at each step to solve the following questions:

(i) Simplify using logarithmic properties,

(ii) Condense the complex logarithm into single term

(iii) Solve: 

Task 3. A research laboratory has been conducting experiments on the rapid increase of cancer cells in an animal. They have observed that cancer cell growth increases by 2% every year with certain medication. Initially, in the year 2018, there were 232.26 units of these cells in the animal.

Using the above data, answer the following questions: 

(i) Create a table to illustrate the yearly increase in cancer cells up to the year 2023.

(ii) Examine the table of values and identify the mathematical function that represents this growth pattern, specifying the key factors of the mathematical function.

(iii) Utilize this mathematical function to project the level of cancer cells in 10 years, assuming the growth rate continues at the same pace

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