how do you find the value of log125 by using log table?
Answers
,Well, you could always use a calculator, computer or a slide rule, as they are not log tables! But, let’s see if there is another way.
As you haven’t specified a base for your logarithms, I’ll do this for you. Let’s use base 5.
log5(125)=log5(53)=3log5(5)=3
That was too easy; let’s try another base, how about using natural logs:
ln(125)=3ln(5)=6arctanh(5−15+1)=6arctanh(23)
I could use a calculator or computer, but let’s see if there is another way. How about a table of inverse hyperbolic tangents? It’s not a log table, so is allowed per your question. But, that’s a boring solution, let’s try another way.
arctanh(23)=∑∞k=012k+1(23)2k+1
=23+13(23)3+15(23)5+17(23)7+...
=23+881+321215+12815309+...
Now, using a calculator, we have
≈0.666667+0.098765+0.026337+0.008361=0.800131
Thus ln(5)≈6×0.8=4.8
As the answer is approximately 4.828314, this is not a brilliant approximation; however, by adding more terms, we can improve it. Adding another four terms in our expansion of arctanh gives us an approximation of 4.827716; rounding this to 3 decimal places gives us 4.828. We need another four terms to give us an answer correct to four decimal places, but just one more to get us to five