How do I find the decimal part of the answer of a logarithm problem

This isn't a homework question, just something I'm curious about, but you can treat it that way if you like. I have been trying to solve logarithm problems with decimal answer.

For example log base 2 of 6 :

$$\log_2 6 = 2.584962500721156$$

Another example log base 3 of 8 :

$$\log_3 8 = 1.8927892607$$

Overall :

$$\log_X Y = 0.000000000$$

It's not about a particular base.

Please tell me how do I find the decimal part of the answers ?

• Since this is a nonlinear equation, and you really want to do it by hand, I would suggest Newton's method on the equation $2^z-6=0$. – Jemmy Nov 22 '13 at 7:03
If you allow yourself a calculator but not Wolfram Alpha, you can use the laws of logarithms: $\log_a(b)=\frac {\log_e(b)}{\log_e(a)}$. Scientific calculators will have a button for $\ln x$
If you are doing it with just pencil and paper, you are in for a lot of work to get many decimals. The best approach I can imagine is successive approximation using square roots and conversion to natural logs. If you know the old pencil and paper approach to calculating square roots you can get within any dyadic fraction you want. So $\log_2 6=2+\log_2 1.5=2.5+\log_2 \frac {1.5}{\sqrt 2}\approx 2.5+\log_2 1.06066$ Now if you know $\ln 2 \approx 0.693147$, this becomes $\log_2 6\approx 2.5+\frac {\ln 1.06066}{0.693147}$ and you can evaluate the numerator using a Taylor's series rather quickly. If you don't know that $\ln 3 \approx 1.0986$ you can use $\ln 3 = 1+ \ln \frac 3e$ and again use the Taylor series for the last.