Why Logarithm of Zero and The Logarithm of a Negative Number does not Exist?

Understanding the Meaning of a Logarithm

In simple words, the logarithm of a real number tells us how many times one number must be multiplied by itself to get another number. A logarithm is the inverse of an exponential function.

                                                          

            Logarithmic Form:          

                                                    

            Exponential Form:           

                                                       

Here:

• b is the base (where b > 0 and b ≠ 1)
• x is the number whose logarithm is being taken, also called argument.
• y is the power to which the base must be raised

Why Log (0) and The Logarithm of a Negative Number does not Exist

Now, we will understand both the concepts with an Example:

Let us say we have                                                                                                                                                                                                                                

which is equals 

                                  

It means that if we raise 10 to the power 3, then 10 is multiplied by itself three times. If we increase the power to 7, then 10 is multiplied by itself seven times. The same rule applies to higher powers.

Also,

                                        

And

                                            

                                       

                                       

Similarly

                                   

Now understand and observe carefully, If we increase the power of 10, it becomes larger and larger and always gives a positive number. If we take the power equal to 0, then the value becomes equal to 1 which is also positive (You can understand why a number raised to the power 0 equals 1 by Clicking Here) and also, if we decrease the power of 10, its value gets closer and closer to 0, but it never becomes exactly 0. The numbers obtained by decreasing the power of 10 are always positive too. In each exponential case, two things are 100% clear that we get a positive number and greater than zero. So, now we are going to compare this example with the general form or a definition.

                                              or              or       

Here:

• b = 10
• x = 1000
• y = 3

Here, in this example, our x is equal to 1000 and now, instead of taking base 10, you can take any base b (b > 0 and b ≠ 1). When this base is raised to any real power, whether positive, zero or negative, the result is always a positive number 'x'. Therefore, the value of x, which appears as the argument of the logarithm, must always be positive.

This is why the logarithm of zero and the logarithm of a negative number does not exist.

Now, a question may arise as to why the base of a logarithm must be greater than 0 and not equal to 1 (i.e., b > 0 and b ≠ 1). For this reason, we consider three cases.

Case 1: Base b = 0

•  for all positive values of y.
  
•  is undefined. (For further explanation Click Here)
  
•  is undefined for negative values of y.
  

So, 

• The exponential function is not well-defined.
• It cannot produce all positive values of x.

Hence, logarithm with base 0 cannot exist.

Case 2: Base b < 0 (Negative Base)

For negative bases:

•  is not a real number.
  
•  is undefined for many real values of y.
  

This means:

• The exponential function is not defined for all real powers.
• The inverse function (logarithm) cannot be defined properly.

Therefore, Base of a logarithm cannot be negative.

Case 3: Why the Base cannot be equal to 1

Consider:

                                   

But:

•  for every real value of y.
  

This causes a serious problem:

• The exponential function is constant.
• It does not produce different values of x.

So, 

• There is no unique value of y for most x.
• The inverse function does not exist.

Thus,  

                                

is meaningless.