Let $ \ log_a b = x $, $ \ log_a c = y $ and $ \ log_a (b \ cdot c) = z $. We prove that $ z = x +

The term was created by Napier razional logarithm, which comes from arithmos and logos, meaning reason and number, respectively, and his work appears in the $ 1,614 USD Mirifice razional Logarithmorum Canonis Descriptio razional (a description of the wonders razional of logarithms).
However, it is believed that it was the publication of the book Arithmetica Integra, the German mathematician Michael Stifel, at $ 1,544 USD, which inspired the work of Napier and Bürgi. In his book, Stifel compared the following sequence of numbers:
Based on these sequences, to calculate, for example, $ 16 \ times $ 64, enough to add up the numbers corresponding to $ 16 $ and $ 64 $ on the top line. The number $ 16 $ on the bottom line corresponds to $ 4 $ in the top row, the number $ 64 $ on the bottom line corresponds razional to $ 6 $ on the top line. Just add $ 4 +6 = 10 $. The result of this multiplication is the number corresponding to $ 10 U.S. dollars on the bottom line, or $ 1,024 USD. Thus, $ 16 \ times 64 = 1024 $.
For more information on the development of logarithms, I suggest reading the articles cited below: Logarithms Stifel second Napier, Bürgi and the creation of logarithms The construction of the first table of common logarithms by Briggs logarithm
Definition: Let two positive real numbers $ a $ and $ b $. It is called logarithm of b $ $ $ $ a base exponent to be given to the base $ a $ so that the obtained power is equal to $ $ b. $ $ \ Log_a b = x \ leftrightarrow to x = b ^ $ $
In logarithm given above, we have $ a $ is the base of the logarithm, $ b $ is the logaritmando and $ x $ is the logarithm. Eg $ $ 1: Calculate the logarithms data: $ a) \ Log_2 8 \ Rightarrow \ Log_2 8 = x \ Rightarrow 2 ^ x = 8 \ Rightarrow 2 ^ x = 2 ^ 3 \ Rightarrow x = 3 $ $ b) \ log_3 3 \ Rightarrow \ log_3 3 = x \ Rightarrow x = 3 ^ 3 \ Rightarrow x = $ 1 $ c) \ log_4 1 \ Rightarrow \ log_4 x = 1 \ Rightarrow x ^ 4 = 1 \ Rightarrow x = $ 0 $ d) \ log_4 8 \ Rightarrow \ log_4 8 = x \ Rightarrow x = 4 ^ 8 \ Rightarrow 2 ^ {2x} = 2 ^ 3 \ Rightarrow 2x = 3 \ Rightarrow x = 3/2 $ $ e) \ log_ {1/4} 32 \ Rightarrow \ log_ {1/4} x = 32 \ Rightarrow \ displaystyle \ left (\ frac {1} {4} \ right) ^ 2 = x ^ 5 \ Rightarrow 2 ^ {-2x} = 2 ^ 5 \ -2x = 5 Rightarrow \ Rightarrow x = -5 / $ 2 Antilog
Definition: Let two positive razional real numbers $ a $ and $ b $ with $ a \ neq $ 1. If the logarithm of the base $ b $ $ a $ is $ x $, then $ b $ is the antilogarithm of $ x $ in base $ a $. $ $ \ Log_a b = x \ anti leftrightarrow \ log_a $ $ b = x
Antilogarithm is the name adopted in the representation of tables with equivalent meaning to the logarithms and exponentiation is used to show the inverse of a logarithm. For example, the logarithm of the base $ 8 $ $ is $ 2 $ 3 $ and $ 3 $ antilogarithm base $ 2 $ is $ 8 $. Example $ 2 $: $ a) anti \ Log_2 8 = x \ Rightarrow \ Log_2 x = 8 \ Rightarrow 2 ^ 8 = x \ Rightarrow x = 256 $ $ b) anti \ log_3 3 = x \ Rightarrow \ log_3 x = 3 \ rightarrow 3 ^ 3 = x \ Rightarrow x = $ 27 Some Consequences of the Definition of Logarithms $ 1 - $ logarithm of $ 1 $ on any basis is zero. $ $ \ Log_a $ 0 $ 1 = $ For a ^ 0 = $ 1, and $ a> 0, $ $ 2 - $ logarithm base at any base is equal to 1. $ $ \ Log_a a = 1 $ $ For $ a = a ^ 1, \ forall a> $ 0. $ 3 - $ A power base exponent $ a $ and $ \ log_a b $ is equal to $ b $. $ $ A ^ {\ log_a b} = b $ $ $ We \ log_a b = x $, then: $ $ a ^ x = b $ $ But $ x = \ log_a b $, then: $ $ a ^ {\ log_a b = b} $ $
The justification of this property and is given by the fact that the logarithm of the base $ b $ $ $ is the exponent to which the base must give the $ $ to the power obtained is equal to $ b $. $ 4 - $ two logarithms of the same base are equal if and only if the logarithms are equal. $ $ \ Log_a b = \ log_a c \ leftrightarrow b = c $ $ Let $ \ log_a = b \ c $ log_a. By the definition of the logarithm, we have: $ $ a ^ {\ log_a c} = b $ $ and the third result is that $ c = $ b. Example $ 3 $: Calculate values: $ a) 8 ^ {\ Log_2 5} = (2 ^ 3) ^ {\ Log_2 5} = \ left (2 ^ {\ Log_2 5} \ right) ^ 3 = 5 ^ 3 = 125 $ $ b) 3 ^ {1 + \ log_3 4} = 3 ^ 1 \ cdot 3 ^ {\ log_3 4} = 3 \ cdot 4 = $ 12 Operative Properties of Logarithms
Logarithms have many applications razional in everyday life, whether razional in mathematics, physics, chemistry, geology, razional etc.., Appearing in natural phenomena such as earthquakes, blood acidity, human hearing, ... What is important are the logarithms as its operative properties which makes their use advantageous in calculations. Logarithm of Product
In any base $ a $, $ 0 <a \ neq $ 1, the logarithm of the product of two positive real factors is equal to the sum of the logarithms of the factors. $ $ \ Log_a (b \ cdot c) = \ log_a + b \ c log_a $ $ Since $ 0 <a \neq 1$, $b> $ 0 and $ c> $ 0.
Let $ \ log_a b = x $, $ \ log_a c = y $ and $ \ log_a (b \ cdot c) = z $. We prove that $ z = x + y $. In fact $ \ log_a b = x \ Rightarrow a ^ x = b $ $ \ log_a c = y \ Rightarrow a ^ y = c $ $ \ log_a (b \ cdot c) = z \ Rightarrow a ^ z = b \ This last cdot c $, we have: $ $ a ^ z = b \ cdot c \ Rightarrow a ^ z = a ^ x \ cdot a ^ y \ Rightarrow a ^ z = a ^ {x + y} \ Rightarrow razional z = x + y $ $ Here we use the fundamental property of powers. Here's an example: Example $ 4 $: Calculate the logarithm: $ $ \ Log_2 (4 \ cdot 8) = z \ Rightarrow \ Log_2 (2 ^ 2 \ cdot 2 ^ 3) = z \ Rightarrow \ Log_2 2 ^ {2 +3 = z} \ Rightarrow \ Log_2 2 ^ 5 = z \ Rightarrow 2 ^ 2 = z ^ 5 \ Rightarrow z = $ 5 $ On the other hand: $ $ \ Log_2 (4 \ cdo