The logarithm is defined as the inverse function of the exponential. Thus, \(log_a b =c \Rightarrow a^c=b\). Since the right hand side will not define b for all values of c when a is negative, it is common to restrict a to the positive numbers. It is also common to not let a=1, for then b must equal one. However, the definition of a logarithm can be extended to negative a if we allow b to be complex. In this case, new and unusual rules and situations arise. In fact, I see nothing wrong with letting a be complex too. This is a perfect example of a function that can be extended, but is commonly restricted for more practical use.

"Since the right hand side will not define b for all values of c when a is negative.", But \((-2)^2=4\) is true. So b is defined, for c=2, when a is negative.

Yes, for that particular value of c, but not for all values of c. And at this point I am still assuming that a, b, and c are real numbers. So while \((-2)^2\) is a real number, \((-2)^{(\frac{1}{2})}\) is not a real number.

a^y=x
Now a can only be zero if,x=0
and a can be 1 only when y=0
So for inclusion of these values,one needs to modify the domain and range respectively.

## Comments

Sort by:

TopNewestwe can also get this from its graph.....

Log in to reply

because if a=1 it has many possibilities

Log in to reply

Does that \( \log_{-2} 4 \) exist? Well, you would believe it is equal to 2. But there's another result. \((-2)^{\log(4)/(\log(2)+i \pi)} = 4\).

That is to say, the logarithm is not uniquely defined at negative values.

Log in to reply

Frankly speaking you can extend logarithm notion further, for example one defines complex logarithm.

The other cases may be indeces (logarithm anologue for \(\mathbb{Z}\setminus p\mathbb{Z}\)) or

p-adiclogarithmsLog in to reply

The logarithm is defined as the inverse function of the exponential. Thus, \(log_a b =c \Rightarrow a^c=b\). Since the right hand side will not define b for all values of c when a is negative, it is common to restrict a to the positive numbers. It is also common to not let a=1, for then b must equal one. However, the definition of a logarithm can be extended to negative a if we allow b to be complex. In this case, new and unusual rules and situations arise. In fact, I see nothing wrong with letting a be complex too. This is a perfect example of a function that can be extended, but is commonly restricted for more practical use.

Log in to reply

"Since the right hand side will not define b for all values of c when a is negative.", But \((-2)^2=4\) is true. So b is defined, for c=2, when a is negative.

Log in to reply

Yes, for that particular value of c, but not for all values of c. And at this point I am still assuming that a, b, and c are real numbers. So while \((-2)^2\) is a real number, \((-2)^{(\frac{1}{2})}\) is not a real number.

Log in to reply

a^y=x Now a can only be zero if,x=0 and a can be 1 only when y=0 So for inclusion of these values,one needs to modify the domain and range respectively.

Log in to reply