Logarithmic Functions

Logarithmic Functions

The representation of the Logarithmic functions as shown;

f(x)=logb(x)                                      

Here b signifies the base of the function.         

                       

Depending upon the base the function can be a decreasing(value of b lies between 0 to 1) function or an increasing(value b is greater than 1 ) function. Logarithmic functions are also the inverse of exponential functions.

Common Logarithmic Function

The logarithmic function with base 10 is called the common logarithmic function and it is denoted by log10 or simply log.

f(x) = log10 x

Natural Logarithmic Function

The logarithmic function to the base e is called the natural logarithmic function and it is denoted by loge.

f(x) = loge x

Logarithmic Functions Properties

Logarithmic Functions have some of the properties that allow you to simplify the logarithms when the input is in the form of product, quotient or the value taken to the power. Some of the properties are listed below.

Product Rule

logb mn = logb m + logb n

Multiply two numbers with the same base, then add the exponents.

Example : log 30 + log 2 = log 60 


Quotient Rule 

logb m/n = logb m – logb n

Divide two numbers with the same base, subtract the exponents.

Example : log8 56 – log8 7 = log8(56/7)=log88 = 1

Power Rule 

Raise an exponential expression to power and multiply the exponents.

Logb mp =p logb m

Example : log 1003 = 3. Log 100 = 3 x 2 = 6

Zero Exponent Rule

loga 1 = 0.

Change of Base Rule

logb (x) = ln x / ln b or logb (x) = log10 x / log10 b

Other Important Rules of Logarithmic Function

  • Logb b = 1 Example : log1010 = 1
  • Logb bx = x Example : log1010x = x
  • blogbx=x. Substitute  y=logbx,it becomes by=x

There are also some of the logarithmic function with fractions. It has a useful property to find the log of a fraction by applying the identities

  • ln(ab)= ln(a)+ln(b)
  • ln(ax) = x ln (a)

We also can have logarithmic function with fractional base.

Consider an example,

By the definition, loga b = y becomes ay = b

$\frac{4}{9}$49  )y =  $\frac{27}{8}$278  

$\frac{2^2}{3^2}$2232  )y$\frac{3^3}{2^3}$3323  

(⅔)2y = (3/2)3