Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

This could be a basic question. But I would like to know steps I should follow to calculate $\log_2\, 0.667$.

EDIT

In an answer I found it says $(0.038 \log_2 0.038) = -0.181$. How this calculation works?

Is it $0.038 \log(0.038) / \log 2$ ?

share|improve this question
3  
What tools do you have? On a calculator something like log(0.667)/log(2) would give you the answer, and it does not even matter whether your calculator uses common or natural logarithms. –  Henry Mar 5 '12 at 14:23
1  
If you want a fully elementary solution, you can always use the expansion of $\log_2$ as a power series... –  Feanor Mar 5 '12 at 14:27
    
Thank you for your suggestions. –  Prasad Rajapaksha Mar 5 '12 at 14:31
1  
this may help. –  David Mitra Mar 5 '12 at 14:36
1  
You are correct. Using the change-of-base formula, $\log_2(0.038)$ can be written as $\frac{\log(0.038)}{\log(2)}$. –  josh Mar 5 '12 at 15:03

2 Answers 2

up vote 5 down vote accepted

For a rough approximation notice that

$0.667 \approx 2/3$

so $\log_2\left( 2/3\right) = \log_2\left( 2\right) - \log_2\left( 3\right) = 1 - \log_2\left( 3\right)$

From here one can use a change of bases. Like in the answers above

$ \log 0.667 \approx 1 - \frac{\ln 3}{\ln 2}$

(For the wise kids we know that $\ln 2 \approx 0.69$ and $\ln 3 \approx 1.1$ so )

$ \log 0.667 \approx 1 - \frac{\ln 3}{\ln 2} \approx 1 - \frac{1.1}{0.69} = -\frac{41}{69} \approx -0.59 $

Which is quite a good approximation

share|improve this answer

That

$\log_a(x) = y$

means by definition that

$a^y = x$

So to find $\log_2(0.667)$ you would (using the definition) need to solve the equation $2^y = 0.667$. We can solve that equation by "taking" $\log = \log_{10}$ on both sides, so we get

$\log(2^y) = \log(0.667) \Rightarrow$

$y\log(2) = \log(0.667) \Rightarrow$

$\log_2(0.667) = y = \frac{\log(0.667)}{\log(2)}$

To evaluate this expression you would need a calculator. Note you could also use the natural logarithm ($\ln = \log_{e}$, $e = 2.718281...$) and get

$\log_2(0.667) = \frac{\ln(0.667)}{\ln(2)}$

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.