How to figure out the log of a number without a calculator? I have seen people look at log (several digit number) and rattle off the first couple of digits. 
I can get the value for small values (aka the popular or easy to know roots), but is there a formula.  Similar to how to tell if a number is divisible by an integer.  
I have read this and this but could some one explain why it works? 
 A: One can get very good approximations by using 
$$\frac 1 2 \log  \left|\frac{1+x}{1-x}\right| =x+\frac {x^3} 3+ \frac {x^5}5+\cdots$$
Say you want to get $\log 3$. Then take $x=1/2$. Then you get
$$\log 3 \approx 2\left( \frac 1 2 +\frac 1 {24} + \frac 1 {140} \right)=1.0976190\dots$$
The real value is $\log 3 = 1.098612289\dots$
Take another term to get
$\log 3 \approx 1.098065476\dots$.
Note that this particular series has the advantage that for $x &lt 1$ (which is where it works) you get "exponentially increasing" approximations.
A: This can be done by recourse to Taylor series. For $\ln(x)$ centered at 1, i.e. where $0 < x \leq 2$:
$$
\ln(x)= \sum_{n=1}^\infty \frac{(x-1)^n}{n}= (x-1) - \frac{1}{2}(x-1)^2 + \frac{1}{3}(x-1)^3 + \frac{1}{4}(x-1)^4 + \cdots 
$$
A: A curve can be approximated with line segments.
If $4 \le x \lt 5$, then $\log(x) \approx 0.1(x+2)$.  For example, $\log(4.1) \approx 0.61$, $\log(4.2) \approx 0.62$, $\log(4.3) \approx 0.63, \ldots, \log(4.9) \approx 0.69$.
If $7 \le x \lt 10$, then $\log(x) \approx 0.1(x/2 + 5)$.  For example, $\log(7.2) \approx 0.1(7.2/2 + 5) \approx 0.86$.
