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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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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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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