Analysis of Logarithmic and Exponential Functions with Examples

This assignment provides a comprehensive overview of logarithmic and exponential functions. It begins by defining logarithmic functions in terms of the inverse of exponential functions, explaining the key characteristics such as the increasing nature of the function, domain, and range. The document then provides examples of logarithmic functions in real-world scenarios, including sound measurement, the Richter scale, and pH values. The assignment then transitions to exponential functions, defining them as functions with a constant raised to a variable exponent, highlighting their rapid increase, and applications in growth and decay scenarios. Examples of exponential functions include compound interest and carbon dating. Finally, the assignment emphasizes the importance of understanding the function's limits, slope, and nature to create accurate graphs. The document references key mathematical resources to support the explanations.

Running head: LOGARITHMIC AND EXPONENTIAL FUNCTIONS 1
Logarithmic and exponential functions
Name:
Institution:

LOGARITHMIC AND EXPONENTIAL FUNCTIONS 2
The explanation is as the following: -
Logarithmic Function
Logarithmic functions can also be defined in terms of inverse of exponential functions. For
instance, the exponential function y = ax can be inversed to x = ay (Beyer, 2018).
Concepts
 One can have logarithmic function by inversing the exponential function.
 It is defined as y = logax. Here a is the base such that a is positive and not equal to 1.
 y = loga .x is denoted as being equal to the exponential equation x = ay (Polyanin &
Nazaikinskii, 2015).
It has some following characteristics:-
 The function always increases as increases.
 The domain is all positive real number except 1.
 The range is all real numbers.
 a= 10 and a = 3 are some standard value of a and find application in many problems.
----> Concepts to accommodate new concepts:-
 The function always increases as x increase.
 Inverse of the exponential function.
 The base is always positive except 1.
----> Example:-
1. Sound (decimal measure)
2. Earthquake (Richter scale)

LOGARITHMIC AND EXPONENTIAL FUNCTIONS 3
3. pH of any liquid.
4. The brightness of the star.
----> Logarithmic function is used mostly where we had to deal with the term with exponent
(Beyer, 2018).
----> To get the graph, first we need to look at the condition of the base, the point at which it
intersects the x-axis, its slope, and its nature.
Exponential function
Concepts
1. Other than one raised to a variable exponent, an exponential function is denoted as a function
with a positive constant (Polyanin & Nazaikinskii, 2015).
2. In this case, constant is raised to some variable exponent.
3. If b is a number such that b ̸ = 1 and b > 0 and, therefore, the exponential function is a function
in kind of f(x) = bx.
Here x is any number and b is termed as the base. Here, the exponent is the independent variable
x and b is fixed. This is precisely reverse to many of the function which you have studied earlier
(Earlier constant is raised to some variable for example: x2).
It has some following characteristics:-
 It increases very rapidly.
 Growth and decay are defined by this type of function.
 The always intersects y-axis at (0,1).
 The domain is all real numbers.

LOGARITHMIC AND EXPONENTIAL FUNCTIONS 4
--> Concepts need to accommodate these new concepts are: -
1. Constant is raised to some variable.
2. The increase is very rapid.
3. Growth and decay function.
---> Example:
 Compound interest: A = P (1 + i)n. Here (1 + i) is the constant while n is the number of periods
which is a variable (Polyanin & Nazaikinskii, 2015).
 Carbon dating: To find how old any tree is.
 Model population.
---> The fact occurring is very clear. Exponential function defines decay and growth (Beyer,
2018).
----> To get a graph, we need to find the lower and upper limit of the function, nature of the
slope and study the nature of the change of slope.

LOGARITHMIC AND EXPONENTIAL FUNCTIONS 5
References
Beyer, W. H. (2018). Handbook of Mathematical Science: 0. CRC press. [Online]. Available
from: https://content.taylorfrancis.com/books/download?dac=C2017-0-66677-
X&isbn=9781351081115&format=googlePreviewPdf
Polyanin, A. D., & Nazaikinskii, V. E. (2015). Handbook of linear partial differential equations
for engineers and scientists. Chapman and hall/crc. [Online]. Available from:
https://www.taylorfrancis.com/books/9780429166464