Logarithm conceptual problem

$f(x)=\log_a x$ is defined only for $a>0$ and $a \neq 1$ . Why is it so? Why is it not defined for negative values of a?

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Comments

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.

Bob Krueger - 8 years ago

"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.

Shubham Srivastava - 8 years ago

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.

Bob Krueger - 8 years ago

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-adic logarithms

Nicolae Sapoval - 8 years ago

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.

George Williams - 8 years ago

because if a=1 it has many possibilities

pelajar mipa - 8 years ago

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

Ajay Kotwal - 7 years ago

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.

Harsa Mitra - 8 years ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$