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I'm trying to figure out how to calculate the base if:

$ \log_b 30 = 0.30290 $

How do I find $b$ ?

I've slaved over the Wikipedia page for logarithms, but I just don't get the mathematical notations.

If someone could let me know the steps to find $b$ in plain english, I'd be eternally grateful!

4 Answers 4

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You need to think about the definitions.

Since $a^b=c$ can be rewritten as $\log_a c = b$.

That should tell you that,

$b^{0.30290} = 30$

and then,

$b = \exp {\frac{\ln 30}{0.30290}} $

  • 9
    After your first equation $b^{0.30290} = 30$, it would IMHO be more straightforward to raise both sides to the power $1/0.30290$ and write $b = 30^{1/0.30290}$. (For those who don't know the $\exp$ and $\ln$ functions but have some middle-school level understanding of exponentiation.)2011-06-20
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The change-of-base identity says the following: fixing $\ln$ to mean the natural logarithm (logarithm with base $e$), $ \log_b x = \frac{\ln x}{\ln b} $ and as a consequence, you can derive the statement that $ \log_b x = \frac{1}{\log_x b}. $

This tells you that your statement $ \log_b 30 = 0.30290 $ is equivalent to $ \log_{30} b = \frac{1}{0.30290}$ so that

$ b = 30^{\frac{1}{0.30290}} \sim 75265.70 $

  • 0
    Use the change of base identity on both sides $ \log_b x = \frac{\ln x}{\ln b} = \left( \frac{\ln b}{\ln x} \right)^{-1} = \frac{1}{\log_x b} $2011-04-07
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Once you have log of one base (e.g. the natural log $\ln$), you can easily calculate the log of any basis via $\log_b a = \frac{\ln a}{\ln b}.$

In your case you want to solve $\log_b a =c$ for $b$, which is easily done using the formula above with the solution $ \ln b = \frac{\ln a}{c}$ or equivalently $b = \exp \left( \frac{\ln a}{c} \right).$

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    Thanks, gu$y$s. I have noticed some exponential (and fraction-al) expressions being tin$y$ in markup. Wait... was I just referred to the wikipedia page for e^x??? :)2011-04-06
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In Excel: to quickly calculate the elusive "e":

To calculate "e" (the base of LN): e = x^(1/LN(x)) Wherein: x = any number >or< 1 but > 0