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I have a short question. How can I calculate a logarithm without a calculator? For example: Log base 8 of 4, log base 16 of 2.. Thanks.

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marked as duplicate by N. F. Taussig, Rohan, user91500, GNUSupporter 8964民主女神 地下教會, hardmath Feb 27 '17 at 20:13

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  • $\begingroup$ Use $\log_b(b^a)=a$ which translates to $\log_{c^n}(c^m)=\dfrac{m}{n}$ $\endgroup$ – Henry Feb 27 '17 at 10:24
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$y = \log_{b} x $ is equivalent to $$ x = b^y$$
hence $y=\log_{8} 4 $ is equivalent to $$4= 8^y$$ $$ 2^2 = 2^{3y}$$ since you have same base then you can set the exponents to be equal. Therefore: $$ 2 =3y$$ so $$y=2/3 $$

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  • $\begingroup$ If you precede the logarithm name with a backslash: \log it will become a LaTeX symbol, rendered in upright font and looking like a function name: $\log$, instead of italic font log → $log$, which looks somewhat like a product of three varables $l\cdot o\cdot g$. $\endgroup$ – CiaPan Feb 27 '17 at 10:39
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for example we have $$\log_8 4=x$$ then we get $$8^x=4$$ and this is equivalent to $$2^{3x}=2^2$$ thus we get $$3x=2$$ or $$x=\frac{2}{3}$$ or write $$\log_{16} 2=x$$ then we get $$16^x=2$$ thus we get $$2^{4x}=2^1$$ and we get $$x=\frac{1}{4}$$

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More general way to do it is as follows

  1. Convert it to the natural logarithm for $N > 1$ $$\log_{10}(N) = \frac{\ln(N)}{\ln(10)}$$
  2. Use the following property$$\ln(N)=-\ln\left(\frac1{N}\right)=-\ln\left(1 - \left(1-\frac1{N}\right)\right)$$
  3. Use Taylor series for $\ln(N)$ up to a required precision $${\displaystyle {\begin{aligned}\ln(1-x)&=-\sum _{n=1}^{\infty }{\frac {x^{n}}{n}}=-x-{\frac {x^{2}}{2}}-{\frac {x^{3}}{3}}-\cdots &&{\text{ for }}|x|<1\\\ln(1+x)&=\sum _{n=1}^{\infty }(-1)^{n+1}{\frac {x^{n}}{n}}=x-{\frac {x^{2}}{2}}+{\frac {x^{3}}{3}}-\cdots &&{\text{ for }}|x|<1\end{aligned}}}$$
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