Applications of exponentials and logarithms
Introduction to logarithms
Introduction to logarithms is the inverse of exponential functions. Considering a simple graph of a single line, we can derive an equation where function (f) of ‘y’ equals to ‘2x’. Considering inverse functions, we can conclude that ‘y’ equals ‘2x’ and ‘x’ equals to ‘2y’. To get the value for y, logarithms are used to solve the equation ( Moore 1998). The result will be; log to the base of 2 of ‘x’ equals ‘y’. ‘y’ usually refers to the power to which one raises ‘2’ to get ‘x’ This can be simplified as follows;
F(y) =2x
y =2x
x=2y (exponential form)
log2 x=y (logarithmic form)
Exponential function notation is used to express functions containing exponents and logs. For example, assume function f(y) equals to logarithms base 4 of ‘x’. Assuming the value of ‘y’ is 16, and then we can get the number to which we can raise 4 to get 16 (Moore 1998). This can be done as follows;
F(y) = log4 x
F (16) = log4 16=x
16=4x
X=2
Applications of logarithms
Logarithms are used to get values of variables in an exponential function. For example, consider the interest formula in general in multiple compounding, there are several variables.
A= P (1+r/n) nt
A= Total Amount (Principal+ Interest)
P= Original Principal
r= interest rate
t= time
n= number of times per year (quarterly, semi-quarterly among others). Logarithms are a powerful tool in banking. If a client wants to know the period it will take for him or her to earn a given amount, then logarithms are used for calculation (Moore 1998).
Logarithms and pH scale. Logarithms are used in the measurement of the concentration of hydroxide (H+) and hydroxyl (OH–) ions in water. For example, at 250C the concentration of H+ is 1.0 *10-7 and that of OH– is 1.0 *10-7 in pure water. pH is defined as the negative logarithm of hydroxide and hydroxyl ions; which is -7. This means logarithms are used to measure the pH of fluids, for instance, water is 7 (Moore 1998).
Decibel (db) is the unit used to measure the intensity of sound. The decibel scale determines the power of sound. The smallest unit measured by the decibel scale is 0 db; which is total silence. According to the decibel scale, a whisper has 15dbs, normal conversation 60 db, and a gunshot has 140 db. The intensity of sound is dependent on the distance from the source of sound (Moore 1998).
The Richter scale measures the magnitude of an earthquake and records it in the amplitude of zigzag waves. The magnitude increases at the scale of 10 times and energy 31 times. The strongest earthquake ever recorded was 9.5 in Chile. Richter scale was invented by Charles in 1935 in California. The Richter scale determines the magnitude of an earthquake by measuring the size of the waves.
Matrices are rectangular arrangements of numbers, expressions, or symbols. An individual item in the matrix is referred to as an element or entry (Moore 1998).
Vectors are the object of geometry which has length and direction. Vectors are represented by a segment of line with a defined direction connecting the point of origin with a terminal point.
Math help notebook is a textbook which deals with the matrix. It has a definition of terms, how to add, subtract, and multiply matrices. It has details on how to solve linear equations (Moore 1998).
The cell and DNA
How the body works the cell
The cell contains proteins and waste which are emitted through the cell membrane. The cell feeds on anything which comes in. food has eaten merges with liposome. The cell is a living thing that makes the proteins that make up the body (Kingston 2001).
The inner life of the cell
It is composed of red blood cells and leukocytes. Red blood cells are carried by the blood to supply oxygen. Leukocytes interact with endothelium walls to form proteins that make up both cells. Cells also contain lipids and cholesterol which enhance their ability to function (Kingston 2001).
Introduction to DNA
DNA is a genetic material inside the human body. DNA replicates in the nucleus of every cell in the body. The replication takes place during the growth of nails and hair. Replication creates two similar or identical cells (Kingston 2001).
The structure of DNA
The DNA is composed of nucleotide which is in two sequences that run along a molecule. The spiral part is not twisted and DNA can be viewed as two strands that run parallel to each other. The linear sequences of DNA form letters A, C, G, and T.
DNA transcription
This is the process through which the information in DNA is copied to messenger RNA to produce proteins. The factors which favor the transcription assemble on DNA (Kingston 2001).
DNA replication
It is the process through which genetic information on DNA is passed on. It involves the unwinding and separation of the two strands of DNA.
References
Kingston, R. (2001). Introduction of DNA into mammalian cells. Current protocols in Molecular biology, 3 (4), 50-75.
Moore, F. (1998).An introduction to the mathematics of digital signal processing: Part I. Algebra, trigonometry, and the most beautiful formula in mathematics. Computer Music Journal, 2 (1), 38-47.