If you have memorized that
log2=0.3
you can follow this way
log5=log(10/2)=1-log2=1-0.3=0.7
If you want a general way to find logarithms without using calculators or tables, you could use this formula:
(1/2)ln|(1+x)/(1-x)|=f(x)=x+x^3/3+x^5/5+...
And
logy=lny/ln10=2/ln10*(1/2*ln|y|) => logy=0.869*(1/2*ln|y|) where y=(1+x)/(1-x)
(Note1: you can use 2/ln10= 0.868589 with the precision you like. Using two terms of the series, 0.869 has a proper level of precision. Note 2: the values of x must be smaller than 1.)
We can't calculate log5 directly because
(x+1)/(1-x)=5 => x+1=5-5x => 6x=4 => x=1.5
And the series doesn't converge when x>1
But since 5=2*2.5
for y_1=2 ->(x+1)/(1-x)=2 => x+1=2-2x => x=1/3~=0.3333
f(x=1/3)=1/3+1/3^3*1/3=1/3+1/81=0.3333+0.0123=0.3456
for y_2=2.5 -> (x+1)/(1-x)=2.5 => x+1=2.5-2.5x => 3.5x=1.5 => x=3/7~=0.4286
Of course we can use this x=0.4286. But perhaps there is an easier way (without a calculator we need to think of this) such as:
Considering that 5=2^2*1.25 (and since we have already calculated f(x=1/3)):
for y_2=1.25 -> (x+1)/(1-x)=1.25 => x+1=1.25-1.25x => 2.25x=0.25 => x=25/225=1/9~=0.1111
f(x=1/9)=0.1111+1/9^3*1/3=0.1111+1/729*1/3=1/9+1/2187=0.1111+0.0005=0.1116
(as to the number 0.0005 just remember that 10/2=5)
Using the results above
log5=0.869[2*(1/2*ln|2|)+(1/2*ln|1.25|)]=0.869[2*f(x=1/3)+f(x=1/9)]=0.869[2*0.3456+0.1116]=0.869[0.6912+0.1116]=0.869*0.8028=0.6976332 or 0.698 in 3 decimals
We should be aware that this last estimate is smaller than the correct result.
(In fact log5=0.6990)
