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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.

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5 Answers 5

up vote 5 down vote accepted

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

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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, $\frac12 = \log_4 2$

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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.

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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 8 '14 at 9:17
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 8 '14 at 9:18
edited looks more clear now. Thanks for the help. Bookmarking MathJax Basic tutorial –  Abhimanyu Aryan Jul 8 '14 at 9:40

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.

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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 $4^{\frac{1}{2}}=2$

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"3 powers to 10" - is that a common way to present this? I would say "10 to the 3rd power" or simply "10 to the 3rd." –  JoeTaxpayer Jul 8 '14 at 16:14

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