Exponent

Extension of logarithms to fractional and negative values

Originally, exponents were natural numbers: it's easy to see the meaning of an expression such as $10^{3}=10\times 10\times 10$. Rules for adding and multiplying exponents were noticed, and to extend the idea to fractions and negative numbers it was assumed that the same rules would apply. To define a meaning for a fractional value such as $10^{\frac {1}{2}}$, consider that, using a rule for multiplying exponents,

$(10^{\frac {1}{2}})^{2}=10^{{\frac {1}{2}}\times 2}=10^{1}=10$

Therefore $10^{\frac {1}{2}}$ must be ${\sqrt {10}}$ and this then supplies a value for $\log _{10}({\sqrt {10}})=0.5$. Values for many other numbers can be worked out similarly using cube roots and so on, and values for all real numbers can then be defined using limits.

To assign meaning to negative values of exponents, note the rule that

$b^{x}b^{y}=b^{x+y}.$

So, for example, to find the meaning of $10^{-3}$, consider

$10^{-3}\times 10^{4}=10^{-3+4}=10^{1}=10$

Therefore,

$10^{-3}={\frac {10}{10^{4}}}=0.001$

and it then follows that

$\log _{10}(0.001)=-3,$

or in general,

$\log _{b}\left({\frac {1}{x}}\right)=-\log _{b}(x).$

By a similar argument it can be established that $b^{0}=1$ for any base $b>1$ and therefore that $\log _{b}(1)=0$.