When the instructions say „extend,“ they mean they gave me a sense of protocol with many things, and they want me to use the protocol rules to break down the log into many separate protocol terms, each containing only one thing in its particular protocol. That is, they gave me a protocol with a complicated argument, and they want me to convert it into several protocols, each with a simple argument. The logarithm of a product is the sum of the logarithms of the factors. Napier died in 1617 and Briggs continued alone, publishing a table of logarithms in 1624 calculated to 14 decimal places for the numbers 1 to 20,000 and 90,000 to 100,000. In 1628, Dutch publisher Adriaan Vlacq published a 10-digit table for values from 1 to 100,000 and added the 70,000 missing values. Both Briggs and Vlacq worked on setting up trigonometric log tables. These first tables were accurate either to one hundredth of a degree or to one minute of arc. In the 18th century, tables for 10-second intervals were published, adapted to tables with seven decimal places. In general, finer intervals are needed to calculate logarithmic functions of smaller numbers, for example when calculating log sin x and log tan x functions.

The invention of logarithms was anticipated by comparing arithmetic and geometric sequences. In a geometric sequence, each concept forms a constant relationship with its successor; Like what. 1/1 000, 1/100, 1/10, 1, 10, 100, 1 000.. has a common ratio of 10. In an arithmetic sequence, each successive term is distinguished by a constant called a common difference. Like what. −3, −2, −1, 0, 1, 2, 3. has a common difference of 1.

Note that a geometric sequence can be written in relation to its common relationship. For the example of a geometric sequence given above: . 10−3, 10−2, 10−1, 100, 101, 102, 103. Multiplying two numbers in the geometric sequence, such as 1/10 and 100, is equivalent to adding the corresponding exponents of the common ratio −1 and 2 to obtain 101 = 10. Thus, multiplication is converted to addition. However, the original comparison between the two series was not based on an explicit use of exponential notation; It was a later development. In 1620, the Swiss mathematician Joost Bürgi published in Prague the first painting based on the concept of relation to geometric and arithmetic sequences. Logarithms were quickly adopted by scientists due to various useful properties that simplified time-consuming and tedious calculations. Specifically, the scientists were able to find the product of two numbers m and n by looking up the logarithm of each number in a special table, adding the logarithms, and then looking at the array again to find the number using that calculated logarithm (known as the antilogarithm). Expressed in general logarithms, this relationship is given by log mn = log m + log n. For example, 100 × 1,000 can be calculated by looking for the logarithms of 100 (2) and 1,000 (3), adding the logarithms (5), and then finding the antilogarithm (100,000) in the table. Similarly, division problems with logarithms are converted into subtraction problems: log m/n = log m − log n.

That`s not all; The calculation of powers and roots can be simplified using logarithms. Logarithms can also be converted between arbitrary positive bases (except that 1 cannot be used as a basis, since all its powers are equal to 1), as shown in the table of logarithmic laws. Only logarithms for numbers between 0 and 10 were generally included in logarithic tables. To obtain the logarithm of a number outside this range, the number was first written in scientific notation as the product of its significant numbers and exponential power – for example, 358 would be written as 3.58 × 102 and 0.0046 as 4.6 × 10−3. Then the logarithm of the significant digits – a decimal fraction between 0 and 1, known as a mantissa – would be found in an array. For example, to find the logarithm of 358, one would have to search log 3.58 ≅ 0.55388. Therefore, log 358 = log 3.58 + log 100 = 0.55388 + 2 = 2.55388. In the example of a number with a negative exponent, for example 0.0046, the logarithm 4.6 ≅ 0.66276 would be searched. Therefore, log 0.0046 = log 4.6 + log 0.001 = 0.66276 − 3 = −2.33724. Since taking a logarithm is the opposite of exponentiation (more precisely, the logarithmic function $log_b x$ is the inverse function of the exponential function $b^x$), we can derive the basic rules for logarithms from the basic rules for exponents. If we take the logarithm on both sides of $e^{ln(xy)} =e^{ln(x)+ln(y)}$, we get $$lnbigl(e^{ln(xy)}bigr) =lnbigl(e^{ln(x)+ln(y)}bigr).$$ Logarithms and exponentials cancel each other out (equation eqref{lnexpinversesb}), which is our product rule for logarithms $$ln(xy) =ln(x)+ln(y).$$ How to take an expression with multiple logarithms and write it as an expression, which contains only one logarithm? Like exponents, logarithms have rules and laws that function in the same way as exponent rules. It is important to note that logarithm laws and rules apply to logarithms of any base.

However, the same basis must be used when making a calculation. Logarithmic expressions can be written in different ways, but under certain laws called logarithm laws. These laws can be applied on any basis, but the same basis is used in a calculation. The video explains and applies various properties of logarithms. The focus is on the application of product properties, quotient and power of logarithms. Let`s look at some of these applications of logarithms: The availability of logarithms greatly influenced the shape of plane and spherical trigonometry. Trigonometry methods have been rewritten to produce formulas in which logarithm-dependent operations are performed at once. The use of tables then involved only two steps, the search for logarithms and, after calculations with logarithms, the search for antilogarithms.