Logarithm

In mathematics, a key function that serves as the inverse of an exponential function is the logarithm function. The basic form of a logarithmic function is:   
  

f(x) = log_a (x) or y = log_a (x)

Here, a > 0 and a ≠ 1. The exponential function of the above log form is:
   a^y = x
Natural logarithm (ln) and common logarithm (log) are the two types of logarithmic functions. 
   

For example, f(x) = ln (x - 2) represents a natural logarithmic function, while g(x) = log (x + 5) -2 represents a common logarithmic function. 

The logarithmic functions help to solve exponential equations, especially when the exponent is not an integer. For instance, 2^x = 10 can be transformed to log₂ (10) = x, and we can easily find the value of x, even if it is not a whole number. 
The formula for converting an exponential function into a logarithmic function is:       

A logarithm indicates the exponent to which base must be raised to get a value inside the log. Logarithms cannot be used for negative numbers, but they can be used for decimals, fractions, and whole numbers.

The logarithm of zero or negative numbers cannot be calculated. Basic logarithmic function is:

f(x) = log (x) (or y = log (x) ) where x>0 


The set of all positive real numbers is known as the domain ( x > 0) or (0, ∞ ). The output value, y, can be any real number, positive, negative, or zero. A list of y-values for various x-values is provided here:

• When x = 1, y = log (1) = 0
   
• When x = 2, y = log (2) ≈ 0.3010 
   
• When x = 0.2, y = log (0.2) ≈  - 0.6990 
   
• When x = 0.01, y = log (0.01) = - 2

The range of every logarithmic function is always a real number, and the domain is always greater than 0. For example, the domain and range of the logarithmic function f(x) = log (x + 3).

The argument of the function should be greater than 0 to determine the domain. 
Now, we can solve for x:
      x + 3 > 0
      x > -3 
Thus, the domain is x > -3 or (-3, ∞).

Next, we can find the range; it can be any real number. Therefore, the range is Range = R. 

Hence, the range of f(x) = R. 

A logarithmic graph depicts how the logarithmic function varies with different values of input x. The domain is the set of positive real numbers, while the range is a set of any real numbers. Concerning the line y = x, the logarithmic and exponential function graphs are symmetrical. It means that the graph of an exponential function across the line y = x is reflected in the graph of the logarithmic function. 

• y = 0 when x = 1 because log_a (1) 0 for any base a. Then the graph has an x-intercept.  

• log_a (0) is not defined, so the graph does not have a y-intercept. 

Take a look at the graphs of exponential and logarithmic functions for a better understanding. 

The key properties of logarithmic graphs are listed below:

• a > 0 and a ≠ 1. It explains that the base must be greater than 0 and it should not be equal to 1. 

• When a > 1, the logarithmic graph increases, while the graph decreases when 0 < a < 1. 

• The input to the function will always have a positive value. 

• The range or the output of a function can be any real number. 

Menemukan titik-titik yang menggambarkan perilaku suatu fungsi dan menggambar kurva melalui titik-titik tersebut dikenal sebagai menggrafik fungsi logaritma. Tergantung pada nilai basis, arah kurva dapat meningkat atau menurun. 

Kurva meningkat jika basis lebih besar dari 1 (basis > 1), dan menurun jika basis berada di antara 0 dan 1 (0 < basis < 1). Untuk menggrafik fungsi logaritma, kita harus mengikuti langkah-langkah tertentu, yaitu:
 

Langkah 1: Identifikasi domain dan range. 

Langkah 2: Temukan asimtot vertikal dengan menetapkan argumen sama dengan 0. Ingat bahwa grafik logaritma memiliki asimtot vertikal tetapi tidak memiliki asimtot horizontal.

Langkah 3: Tetapkan argumen sama dengan 1 dengan mensubstitusi nilai x. Untuk menemukan titik potong-x, gunakan sifat log_a (1) = 0. 

Langkah 4: Tetapkan argumen sama dengan basis dengan mensubstitusi nilai x. Untuk menemukan titik lain pada grafik, gunakan sifat log_a (a) = 1. 

Langkah 5: Gambar kurva dengan menghubungkan kedua titik dan memperpanjang kurva menuju asimtot vertikal.

Mari kita lihat contoh ini untuk membuatnya lebih mudah dipahami. 
Grafik fungsi logaritma f(x) = log₂ (x - 1)

Langkah 1: Bentuk dasar fungsi logaritma adalah f(x) = log_a (x). 
Dimana a adalah basis, maka, basisnya adalah 2, dan karena 2 > 1, kurva akan meningkat.  

Langkah 2: Sekarang, tetapkan argumen lebih besar dari 0. 
   x - 1 > 0 
   x > 1 
Jadi, domain = (1, ∞ ). 
Range = R

Langkah 3: Temukan asimtot vertikal dengan menetapkan argumen sama dengan 0. Dalam fungsi logaritma, argumen harus positif ( > 0). 
     x - 1 = 0 
     x = 1 
Maka, asimtot vertikal berada di x = 1. 

Langkah 4: Selanjutnya, kita dapat menemukan titik-titiknya. 
Pada x = 2:
     f(2) = log₂ (2 - 1) = log₂ (1) = 0 
Jadi titiknya adalah (2, 0)

Pada x = 3: 
     f(3) = log₂ (3 - 1) = log₂ (2) = 1 
Maka, titiknya adalah (3, 1)

Langkah 5: Gambar grafik dengan menghubungkan titik-titik, dimulai dari dekat asimtot x = 1. 

Di sini, kurva melewati titik (2, 0) dan (3, 1), dan meningkat secara perlahan. Garis merah menunjukkan asimtot vertikal di x = 1. 

The key properties of logarithmic functions are useful when working with exponents and solving equations involving logarithms. 

• Multiplication property: log_b (a × b) = log_b (a) + log_b (b) 
  When two numbers are multiplied inside a logarithm, you can separate them into a sum of two logs. 

• Division property: log_b (a/b) = log_b (a) - log_b (b)
  Dividing two numbers inside a logarithm is equivalent to subtracting the logarithm of the denominator from the numerator. 

• Change of base rule: log_b (a) = log_c (a) / log_c (b)
  You can change the base by dividing the logarithm of the number by the logarithm of the new base. 

• Power property: log_b (a^x) = x log_b (a) 
  To remove the exponent inside a logarithm, multiply the logarithm by that exponent. 

• Logarithm of 1: log_b (1) = 0 
  Regardless of the base, the logarithm of 1 is always 0. 

• Logarithm of the base: log_b (b) = 1
  If the base and number inside the logarithm are the same, the result is always 1.

A logarithmic function’s derivative explains how the function varies with changes in its input. The reverse of the derivative is the integral of the logarithmic function, which helps to find the original function from its rate of change. The derivative formula for the common and natural logarithmic functions are:

• The derivative of the natural logarithm (ln x):
  d / dx [ln x] = 1 / x  

• The derivative of logarithm with base a (log_a x):
  d / dx (log_a x) = 1 / (x ln a)

The integral formulas of logarithmic functions are: 

• The integral of the natural logarithm (ln x): 
  ∫ ln x dx = x (ln x - 1) + C 

This indicates that when we integrate ln x, we get a formula that includes x, in x, and a constant C. The accumulated area under the curve of ln x is represented by the constant C. 

• The integral of the common logarithm (log x):
  ∫ log₁₀ x dx = x (log₁₀ x - 1) + C 

We get a formula that includes x, log x, and a constant C when we integrate log x. The area under the curve of the common logarithm is represented by the constant C.  

Learning the concept of logarithmic functions helps us to apply them to various real-life situations. Here are some real-world applications of logarithmic functions: 

• Logarithmic functions are used by seismologists to measure seismic waves and analyze earthquakes. They measure the magnitude of earthquakes using the Richter scale, which represents large numbers using logarithmic functions. 

• In scientific laboratories, scientists use the pH scale, which is based on a logarithmic scale, to measure the acidity of various solutions. For example, a solution with pH 4 is ten times more acidic than a solution with pH 5. 

• Shareholders and investors can use the logarithmic functions to calculate their compound interest and total investment rate. For instance, they can calculate the time it takes for an investment to reach a certain amount with a fixed interest rate.