Overview:
Logarithms are useful shorthand to express exponential values. They have many common applications in mathematics, science, and engineering.
What Is the Relationship Between Logarithms and Exponents?
The number 10,000 is equal in exponential form to 104, where 10 is the base and 4 is the exponent. In logarithmic form, log10 10,000 is equal to 4. Similarly, log10100,000 is equal to 5. Many bases can be used in deriving logarithms, as the base can be any positive number that is greater than 1. For example log2 4 is equal to 2, because 22 equals 4, and log2 8 is equal to 3.
What Are Some Properties of Logarithms?
One of the most useful properties of logarithms is that logarithms can be multiplied by adding them. This makes calculations much less complex. For example, log2 32 is equal to 5 and log2 128 is equal to 7. What is log2 of 4096? The number 4096 is equal to 32∙128. Another way to express this relationship is that log2 32 + log2128 is equal to log2 4096. This is correct because log2 4096 is equal to 12, and so is 5 + 7. Similarly, logarithms can be divided by subtracting them, For example, log10 of 1000 is 3 and log10 of 100 is 2. The number 1000 divided by 100 is 10, and 103/102 is also 10 or 101. By definition, log10 10 is equal to 1 or 3-2.
What Are the Natural and Common Logarithms?
The natural logarithm has a base that is an irrational number roughly equal to 2.71828. It is useful in many mathematical relationships, such as growth rate. It is represented on many calculators by the abbreviation ln, The common logarithm is the base10 logarithm. It is often represented as log without a subscript. The numerical system that we use for counting is base 10.
How Are Logarithms Used?
Logarithms are often used in applied calculations in many branches of science. The Richter scale, used to measure the strength of earthquakes, is a logarithmic scale. An earthquake of magnitude 4 is 10 times stronger than an earthquake of magnitude 3 and 1000 times stronger than magnitude 1. The pH value of a liquid in chemistry is also measured on a logarithmic scale.
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