2

How is $$log_42= \frac{1}{2}$$ ?

Any formula to how we calculate this?

I know i am confused when base is larger digit than log value term.

Abhimanyu Aryan
  • 55
  • 4

5 Answers5

7

The $\log$ function is the inverse function to the exponential function. Thus, the number $x=\log_a b$ is the number that solves the equation $a^x = b$.

Apply this to your example: what is $x=\log_4 2$? To what power must you put $4$ to get $2$? Well, you know that $\sqrt 4 = 2$, right? Well, since $\sqrt a = a^{\frac12}$, this means that $4^{\frac12}=2$, and by definition, $\frac12 = \log_4 2$

5xum
  • 123,496
  • 6
  • 128
  • 204
5

In general $$\log_{g}(a)=\frac{\log(a)}{\log(g)}.$$ So $\log_4(2)=\frac{\log(2)}{\log(4)}=\frac{\log(2)}{2\log(2)}=\frac{1}{2}$.

Hakim
  • 10,213
Nicky Hekster
  • 49,281
5

The question "what is $\log 2$ to base 4?" is equivalent to the question "what power of 4 is equal to 2?", by the definition of what a logarithm to a base means.

Thus, you just have to ask yourself what number we need to insert into this:

$$4^w = 2$$ to make it work.

Old John
  • 19,569
  • 3
  • 59
  • 113
  • How do write that little base after log. I don't know how to express mathematical sign on this site. Can you post me link where i can read it. So, people can understand my question better – Abhimanyu Aryan Jul 08 '14 at 09:17
  • 1
    In general, to get subscript this in MathJax, you need to put the expression in between dollar signs and use an underscore to get the subscript. There is a MathJax tutorial here. – Old John Jul 08 '14 at 09:18
  • 1
    edited looks more clear now. Thanks for the help.

    Bookmarking MathJax Basic tutorial

     – Abhimanyu Aryan Jul 08 '14 at 09:40
3

Just to note that here you can take the obvious equation $4=2^2$ and take logs to base $4$ so that $$\log_4 4 = 2\log_4 2$$ or $$2\log_4 2=1$$

This is, of course, wholly equivalent to what others have said, and is not a general formula - but as a means of practical calculation e.g. in an exam under pressure - it could help you to avoid mistakes.

Mark Bennet
  • 100,194
0

I have answered the same kind of question earlier, I am just going to paste the same content here.

Log basically evaluates the exponent(e) wrt to a base (b).

For example, $log_{10} 1000=3$ base being 10. That is, $10$ to the $3rd$ power and you will reach 1000.

So Generally, $log_b(value)=e$ such that, $b^e=value$

If we look at your question and apply the above, then

$log_4 2=\frac{1}{2}$

The base here is 4. Now, 4 to how many powers, so you will reach 2? The answer is $\frac{1}{2}$ as $4^{\frac{1}{2}}=2$

MonK
  • 1,794