According to Napier, logarithms express ratios: formula_7 has the same ratio to formula_8, as formula_9 to formula_10 if the difference of their logarithms matches. Mathematically: formula_11. At first, base e was used (even though the number had not been named yet). Henry Briggs proposed to use 10 as a base for logarithms, such logarithms are very useful in astronomy.
A logarithm tells what exponent (or power) is needed to make a certain number, so logarithms are the inverse (opposite) of exponentiation.
Just as an exponential function has three parts, a logarithm has three parts as well: a base, an argument and an answer (also called power).
The following is an example of an exponential function:
In this function, the base is 2, the argument is 3 and the answer is 8.
This exponential equation has an inverse, its logarithmic equation:
In this equation, the base is 2, the argument is 8 and the answer is 3.
Addition has one inverse operation: the subtraction. Also, multiplication has one inverse operation: the division. However, exponentiation actually has "two" inverse operations: root and logarithm. The reason why this is the case has to do with the fact that the exponentiation is not commutative.
The following example illustrates this:
Logarithms can make multiplication and division of large numbers easier, because adding logarithms is the same as multiplying, and subtracting logarithms is the same as dividing.
Before calculators became popular and common, people used logarithm tables in books to multiply and divide. The same information in a logarithm table was available on a slide rule, a tool with logarithms written on it.
Aside from computations, logarithm also has many other real-life applications:
Logarithms to base 10 are called common logarithms. They are usually written without the base. For example:
This is true because:
Logarithms to base "e" are called natural logarithms. The number e is nearly 2.71828, and is also called the Eulerian constant after the mathematician Leonhard Euler.
The natural logarithms can take the symbols formula_19 or formula_20. Some authors prefer the use of natural logarithms as formula_21, but usually mention this on preface pages.
Logarithms have many properties. For example:
This property is straight from the definition of a logarithm:
The logarithm to base "b" of a number "a", is the same as the logarithm of "a" divided by the logarithm of "b". That is,
For example, let "a" be 6 and "b" be 2. With calculators we can show that this is true (or at least very close):
The results above had a small error, but this was due to the rounding of numbers.
Since it is hard to picture the natural logarithm, we find that, in terms of a base-ten logarithm:
Logarithms which multiply inside their argument can be changed as follows:
For example,
Similarly, logarithm which divides inside the argument can be turned into a difference of logarithm (because it is the "inverse operation" of multiplication):
Before electronic computers, logarithms were used every day by scientists. Logarithms helped scientists and engineers in many fields such as astronomy.
Before computers, the "table of logarithms" was an important tool. In 1617, Henry Briggs printed the first logarithm table. This was soon after Napier's basic invention. Later, people made tables with better scope and precision. These tables listed the values of log("x") and "b" for any number "x" in a certain range, at a certain precision, for a certain base "b" (usually "b" = 10). For example, Briggs' first table contained the common logarithms of all integers in the range 1–1000, with a precision of 8 digits.
Since the function is the inverse function of log("x"), it has been called the antilogarithm. People used these tables to multiply and divide numbers. For example, a user looked up the logarithm in the table for each of two positive numbers. Adding the numbers from the table would give the logarithm of the product. The antilogarithm feature of the table would then find the product based on its logarithm.
For manual calculations that need precision, performing the lookups of the two logarithms, calculating their sum or difference, and looking up the antilogarithm is much faster than performing the multiplication by earlier ways.
Many logarithm tables give logarithms by separately providing the characteristic and mantissa of "x", that is to say, the integer part and the fractional part of log("x"). The characteristic of is one plus the characteristic of "x", and their significands are the same. This extends the scope of logarithm tables: given a table listing log("x") for all integers "x" ranging from 1 to 1000, the logarithm of 3542 is approximated by
Another critical application was the slide rule, a pair of logarithmically divided scales used for calculation, as illustrated here:
Numbers are marked on sliding scales at distances proportional to the differences between their logarithms. Sliding the upper scale appropriately amounts to mechanically adding logarithms. For example, adding the distance from 1 to 2 on the lower scale to the distance from 1 to 3 on the upper scale yields a product of 6, which is read off at the lower part. Many engineers and scientists used slide rules until the 1970s. Scientists can work faster using a slide rule than using a logarithm table.
William Penn
William Penn (14 October 1644-30 July 1718) was an English colonial leader. He was very wealthy and owned slaves who he forced to work on his plantation. He was given what would become the US state of Pennsylvania by King Charles II as a debt to his father. Penn was a member of the Religious Society of Friends, who are known as Quakers.
Penn and his wife were made honorary United States citizens in 1984 by US President Ronald Reagan. He was friends with George Fox.
Babe Ruth
George Herman "Babe" Ruth, Jr. (February 6, 1895 – August 16, 1948) was a famous baseball player during the 1910s and 1930s in Major League Baseball. He played with the Boston Red Sox, New York Yankees, and the Boston Braves, and hit 714 home runs in his career. Only two players, Hank Aaron and Barry Bonds, have hit more. At the beginning of his career, he was a pitcher. He is widely considered the greatest baseball player of all time.
Babe Ruth has been called by many nicknames. Some of the most common are: "The Great Bambino", "The Sultan of Swat", "The of Clout", "The Titan of Terror", and "The King Of Crash" .
Ruth learned to play baseball while growing up in Baltimore, Maryland.His first Major League Baseball (MLB) team was the Boston Red Sox. Ruth began playing as a pitcher. He had some of the best pitching statistics in baseball. The Red Sox won the World Series in 1915, 1916, and 1918.
At that time, there was no designated hitter rule in the American League, where the Red Sox played, so Ruth got chances to hit as a pitcher. The team realized that he was also good at hitting. In 1918, Ruth began hitting more and pitching less. Later in his career, he became an outfielder.
Ruth was becoming a star player. However, by 1919, Red Sox owner Harry Frazee was having problems with money. In 1920, the Red Sox sold Ruth to the New York Yankees for cash. Even though the Red Sox had won several World Series in the years before this, they would not win another one until 2004. Many baseball fans believed that the Red Sox had become "cursed" by trading Ruth, and called this the "Curse of the Bambino". (When the Red Sox finally did win a World Series in 2004, they beat the Yankees in the American League Championship to get there.)
Ruth spent most of the rest of his career with the Yankees, where he became one of the most famous players in baseball history. Ruth helped the Yankees win World Series championships in 1923, 1927, 1928, and 1932. He left the Yankees after the 1934 season and played one last season with the Boston Braves in 1935.