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I have a question, I have a calculator that does $\log$ but I think it does it it in a base ten format for example $\log_{10}(100)=2$ I am wondering how I can solve $\log$ using a base of 2 for example I know $2^7$ is $128$

so $\log_2(128)=7$

Is there any way find $\log$ using base of 2 by hand or some calculator method say I want to find what is $\log(100)$ using a base of two. How can I figure it out because my calculator a ti-83 does not let me.

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Use the change of base formula. –  David Mitra Feb 10 '13 at 18:41
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Also, I assume you meant to write "$\ln(100)=2$"? –  Zev Chonoles Feb 10 '13 at 18:42
    
That's weird; the TI-83 has both an $\ln$ key and a $\log$ key, and the inverse functions $\mathrm e^x$ and $10^x$ indicated above the corresponding keys strongly suggest that, as one would expect, the $\ln$ key computes the natural logarithm and the $\log$ key computes the decimal logarithm. Are you sure you're pressing the $\ln$ key and not the $\log$ key? –  joriki Feb 10 '13 at 18:46
    
Note that $ln$ usually means the natural logarithm with base $e$, and the best way to write logarithms with other bases is to use a subscript such as $log_2$ or $log_{10}$ –  Shard Feb 10 '13 at 18:47
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Notation alert! The "n" in "ln" means specifically that it's the natural logarithm. So writing $\ln(128)=7$ or $\ln(100)=2$ is never correct. In contrast "log" means the logarithm with an implicit base, which can be $2$, $e$ or $10$ depending on the context (different fields have different conventions). –  Henning Makholm Feb 10 '13 at 18:58
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2 Answers 2

up vote 3 down vote accepted

$$ \log_2 x = \frac{\ln x}{\ln 2} = \frac{\log_{10} x}{\log_{10} 2}. $$ This is sometimes called the "change-of-base formula".

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thanks I did not even know about that formula. –  Fernando Martinez Feb 10 '13 at 18:50
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Yes, just use the fact that if you have a logarithm in base $b, \log_b(x)$, then you can convert it to base $a$ by $\frac{\log_b(x)}{\log_b(a)}$.

So, no matter what base your calculator uses, just divide $\ln(128)$ by $\ln(2)$, and you should get 7.

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