# logarithms

• Jul 6th 2011, 03:27 PM
Nemesis
logarithms
Why is it that tha base of a logarithm can't be 1? I can't see anything wrong with this.
Thank you
• Jul 6th 2011, 03:33 PM
e^(i*pi)
Re: logarithms
The basic definition of a logarithm is $\log_b(c) = a \Longleftrightarrow c = b^a$

If we say that $b=1$ then it follows that $c = 1^a$. Yet the laws of exponents say that if we raise 1 to any real number we get an answer of 1.
• Jul 6th 2011, 03:42 PM
Also sprach Zarathustra
Re: logarithms
Quote:

Originally Posted by Nemesis
Why is it that tha base of a logarithm can't be 1? I can't see anything wrong with this.
Thank you

Hint:

$\log_ba=x \Leftrightarrow b^x=a$

$a>0$ and $b>0$
• Jul 7th 2011, 12:17 PM
HallsofIvy
Re: logarithms
Another way of thinking about it is that " $log_a(x)$" is the inverse function to $f(x)= a^x$. But if a=1, $f(x)= 1^x= 1$, as e^(i*pi) pointed out. Because that function, unlike $a^x$ for any other (positive) a, is not "one-to-one", it does not have an inverse.