Limits of Exponential and Logarithmic Functions

Exponential and Logarithmic Functions is an essential concept in Mathematics and Statistics. It is used in various real-world scenarios, including investments, population, and more. The limits of Exponential and Logarithmic functions are utilized in different branches of science and finance. 

What is a logarithmic Function?

The inverses of exponential functions are called logarithmic functions. logax=y is defined to be the inverse of y=ax. The logarithmic function has meaning only when x>0 and  a>0, a not equal to 1. The range of the function extends from -∞  to ∞. It is called the logarithmic function with base a.

Furthermore, if the base of a log is 10, then it is called a common logarithm. A logarithm with the command ‘e’ is called a natural log.  Also, It is important to note that the logarithm of 0 is not defined since one integer multiplied by another number never yields 0.

Example of a logarithmic function

• 72 = 49

Here, if you convert the value in logarithmic function, then base = 7, exponent = 2 and argument = 49. The logarithmic function would be log7 49 = 2

What is an Exponential Function?

An exponential function is the basic power function ax with x in the power. A special case of this which has varied applications in maths is ex, where e is napiers constant and is irrational. The inverse of this function is the natural log function where base is e. It is represented by ln x.

Example of an exponential function

The exponential function of log2 1024 = 10 will be 210  = 1024.

Properties of Exponential and Logarithmic Functions

The following are some of the important properties of exponential functions:

• The set of all real numbers, i.e. R, is the domain of the exponential function
• The exponential function’s range is defined as all positive real numbers
• Because it supports the fact that b0 = 1, for any natural number b > 1, the point (0, 1) is always on the graph of the supplied exponential function
• The exponential function is continually rising; the graph climbs upwards as we move from left to right
• The exponential function is close to 0 for a broad set of negative x values

The essential properties of logarithmic functions are –

• A meaningful definition of the logarithm cannot be derived for negative numbers
• As a result, the log function’s domain is the set of positive real numbers or R+
• The log function’s range is the set of all real numbers
• On the graph of the log function, the point (1, 0) is always present
• The log function is ever-increasing, which means that the graph rises as we walk from left to right
• The value of log x can be made smaller than any given real integer for x values close to zero
• The graph approaches but never meets the y-axis in quadrant IV

Limits of Exponential and Logarithmic functions

Limit Laws for exponential functions

limxex = ;

limx – ex = 0;

limx e-x = 0;

limx -e-x = ;

These laws were created to prove that if an exponential exponent goes to infinity in the limit, the exponential function will also go to infinity. In the same way, if the exponent flows to minus infinity in the limit, the exponential will flow to zero.

Limit Laws for Logarithmic functions

The right-handed limit was operated for limx0+ ln x = -. Because we can’t use negative x’s in a logarithm function, the right-handed limit was used. As x from both the right and left sides of the point in issue should be assessed, whereas x’s to the left of zero are negative, the standard limit cannot exist. We can see that if a log’s argument goes to zero from the right (i.e., it is always positive), the record will go to negative infinity in the limit. In contrast, if the argument goes to infinity, the log will also go to infinity in the limit. Because we can’t plug negative values into the logarithm, we can’t look at a logarithm’s limit as x approaches minus infinity.

Rules of Exponential and Logarithmic Functions.

Derivatives of Exponential and Logarithmic Functions

The formulas for the derivatives of exponential and logarithmic functions are listed below.

With respect to x, ex derivative is stated as:

ddx(ex) = ex

The log x derivative about x is written as follows:

ddx(log x) = 1x

We can obtain different formulas by applying exponential and logarithmic laws to these two formulas.

ddx(eax) = aeax  is the derivative of eax about x.

The nth derivative of eax for x is ddx(eax) = an eax 

Similarly, several formulas for the derivative of exponential functions can be found.

Limits of Exponential and Logarithmic examples 

Example 1

Evaluate the derivative y = e^x^4

Solution 

According to the given function, i.e, y = e^x^4

The derivative would be for x 

ddx= ddx( e^x^4) =  e^x^4 ddx(x4) = e^x^4 4×4-1 = 4x3e^x^4

Example 2

x1xx-1x lnx

Solution 

Step 1 – convert exponential to logarithmic

Takin p=xx. Since x->1, p->1

p1p-1ln p

Step 2 – Transform the limit of Function into known form.

If p 1, then p – 1 0.  Therefore,

p-1 0p-1ln p

Now, taking k = p-1, Hence, p = k+1

k 1kln (k+1)

This mathematical expression is identical to the logarithmic function’s limit rule, but it must be written in reciprocal form.

k 01ln k+1k

Now, Using the reciprocal limit rule, 

1k 0ln k+1k

Evaluating limit of log function, 

=1

Example 3

Simplify: y = 135/133

Solution:

y = 135/133

Using the quotient rule,

y = 135/133 = 135-3

= 132

= 169

Conclusion

This article showcases the limits of Logarithmic and Exponential Functions with its examples. Both these functions have a significant role in various industries and help improve and calculate complex problems easily.