Limits of Logarithmic Functions

Introductions to Limits of Logarithmic Functions :
In mathematics, Logarithm is abbreviations of log functions. The logarithmic function contains three parts such as number, base, the logarithmic itself. This limits of logarithmic functions to discover logarithmic functions and their properties, such as domain, range, x and y intercepts and vertical asymptote. This function is called as the limits of logarithmic function. In this article we shall discuss about limits of logarithmic functions.

Definitions:

The logarithmic functions is called as inverse of exponential functions.

bN = A   is equivalent to N=logbA

Where ,        N = number functions,

A = logarithmic functions,

b = Base functions.

Properties for Limits of Logarithmic Functions:
1. Product functions:

`Log_x AB = log_xA+ log_xB`

2. Quotient functions   

`log_x(A/B)= log_xA - log_xB`

3. Power functions           

.`log_xA^B = B log_xA`

4.`LogBA xx logAB=1`

5.`Log_10(1)=0`

6. `Log_BB=1`

7.  `A ^( log_xA)` =x

Limits of Exponential Functions:

`lim_(x->0^+)`    loga x = `-oo` .    if a > 1

`lim_(x->0^+)`  loga x =  `oo`          if a < 1

`lim_(x->oo)` loga x=  `oo`           if a > 1

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Limits of Logarithmic Functions - Problems:

Limits of logarithmic functions - problem 1:

Solve the given limit function `lim_(x->0)` `log(1 + x^3)/cosx` .

Solution:

Given limit function is  `lim_(x->0)` `log(1 + x^3)/cosx` .

= `lim_(x->0)` `log(1 + x^3)/cosx` .

Apply the limit values in the above logarithmic function,   So, we get

=  `log(1 + 0^3)/cos0`.

we know the value of    03   and   cos 0

03   =  0    and   cos 0 = 1

So,                            =  `log(1 + 0)/1`.

= log 1

=  1 .

Answer:   1 .                              

Limits of logarithmic functions - problem 2:     

Solve the given limit function `lim_(x->0)` `log(5 - x^5)cosx` .

Solution:

Given limit function is  `lim_(x->0)` `log(5 - x^5)cosx` .

= `lim_(x->0)` `log(5 - x^5)cosx` .

Apply the limit values in the above logarithmic function,   So, we get

=  `log(5 + 0^5)cos0`.

we know the value of    05   and   cos 0

05   =  0    and   cos 0 = 1

So,                            =  `log(5 + 0)1`.

= log 5

Answer:   0.6989 .