Logarithms are important in advanced algebra, but can be misunderstood. They involve finding the exponent of a base number to equal a total. Logarithms can be positive, negative, or fractional, and can have different base values. Understanding logarithms is important in many fields, including music theory and psychology.
A logarithm is a mathematical term that can also mean “exponent.” As a basic algebraic concept, it is important to understand how to calculate logarithms for almost any math class involving advanced algebra. Perhaps because the formulation of logarithm problems is somewhat upside down, this particular mathematical concept is very easy to misunderstand.
To understand what a logarithm is, you first need to know what an exponent is. An exponent is a number written in superscript above a base number, such as 23, that indicates how many times the base should be multiplied by itself. This can alternatively be written as “power two to the third”. To calculate the total of 23, simply multiply 2 x 2 x 2 to get 8. So 23=8.
To calculate a logarithm base, a person needs two variables: the base number (2) and the total (8). When looking for a logarithm, what is asked is “Which exponent of 2 equals 8?” or “What power of 2 is 8?” In equation form, this is usually written as log28. Since two must be raised to the third power to equal eight, the answer to this question is written as log28=3.
A logarithm or power need not always be a positive integer. It can also be decimals or fractions, or even a negative number. Log164=5, because 16.5=4. Negative powers require an understanding of how to calculate the inverse of a positive exponent. To calculate a negative logarithm, change it to a positive number, calculate the positive calculation, and then divide one by the answer. For example, to figure out what equals 5-2, find 52=25, then divide 1/25 to get .04, so log5.04= -2.
There are two main types of logarithms that commonly emerge. Base 10 logarithms, which include all of the above examples, are usually written as “log”. However, not all equations are based on base 10, which means that numbers can have different values depending on which base is used. While base 10 is by far the most commonly used type of value system, another form that often occurs in advanced math and algebraic calculations is called a base e, which uses the value of 2.718281828 as the base number. Logarithms that use the base e are referred to as natural logarithms and are usually written as ln instead of log.
Understanding the base function of a logarithm is of paramount importance for advanced math calculations. Logarithms appear all over the place in a variety of astonishing areas of study. While unsurprisingly they play a role in fractal geometry, statistics, and probability functions, they are also sometimes used in fields as broad as music theory and even psychology.
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