How do you evaluate log_2 (1/2)?

Aug 24, 2016

${\log}_{2} \left(\frac{1}{2}\right) = - 1$

Explanation:

${\log}_{\textcolor{red}{2}} \textcolor{b l u e}{\left(\frac{1}{2}\right)}$

Written in this form, it helps to read it as a question ...

"What power of $\textcolor{red}{2}$ will give color(blue)(1/2)?

We should realise that $\textcolor{b l u e}{\frac{1}{2}}$ is the reciprocal of $\textcolor{red}{2}$, and it can be written as ${2}^{-} 1$

Log form and index form are inter-changeable.

${\log}_{a} b = c \Leftrightarrow {a}^{c} = b$

So, ${\log}_{\textcolor{red}{2}} \textcolor{b l u e}{\left(\frac{1}{2}\right)} = \textcolor{t e a l}{- 1} \text{ "hArr " } {\textcolor{red}{2}}^{\textcolor{t e a l}{- 1}} = \textcolor{b l u e}{\frac{1}{2}}$

In the same way, can you answer the following instantly?

log_3 27 ,color(white)(xx) log_5 625,color(white)(xx) log_7 49, color(white)(xx)log_2 32, color(white)(xx)log_8 2?

In reverse order the answers are:

$\frac{1}{3} , \textcolor{w h i t e}{\times \times} 5 , \textcolor{w h i t e}{\times \times} 2 , \textcolor{w h i t e}{\times \times} 4 , \textcolor{w h i t e}{\times \times} 3$

Aug 25, 2016

-1

Explanation:

Let ${\log}_{2} \left(\frac{1}{2}\right) = x$.......................(1)

Another way of writing this is:

${2}^{x} = \frac{1}{2}$.......................................(2)
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
AS an example compare to ${5}^{3} / {5}^{4}$

This is ${5}^{3 - 4} = {5}^{- 1} = \frac{1}{5}$

So any value written as say $\textcolor{red}{{z}^{- 1}}$ is the same as $\frac{1}{z}$
'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

So by comparing to the example:

${2}^{x} = \frac{1}{2} \to x = - 1$

$\textcolor{red}{{2}^{- 1}} = \frac{1}{2}$ as in equation(2)

Thus equation(1) is

${\log}_{2} \left(\frac{1}{2}\right) = - 1$