Solving this problem by hand could be a formidable task depending upon the method of solution. Let's assume we are only restricted from directly using a calculator and we are allowed to use tables, and we can check our progress with a calculator. Furthermore we'll assume that we want a relatively easy way to do this, but we are willing to need to do a lot of handwork. First we do some algebra:
Next we consult logarithm tables by searching for what value of log(x) is closest to 0.1. A careful search puts us between 1.25 and 1.26. That is to say: log(1.25)=0.09691 and log(1.26)=0.1003705
Averaging these two results yields: log(1.255)=0.09864025. The actual value is log(1.255)=0.0986437258, so that's not too bad!
From this we deduce that 0.09864025 < log(x) < 0.1003705, which tells us that log(1.255) < log(x) < log(1.26).
By repeating this process we can muddle our way to a decent approximation.
The value is not rational but here are 20 decimals: log(x)=1.2589254117941672104
Next we consult logarithm tables by searching for what value of log(x) is closest to 0.1. A careful search puts us between 1.25 and 1.26. That is to say: log(1.25)=0.09691 and log(1.26)=0.1003705
Averaging these two results yields: log(1.255)=0.09864025. The actual value is log(1.255)=0.0986437258, so that's not too bad!
From this we deduce that 0.09864025 < log(x) < 0.1003705, which tells us that log(1.255) < log(x) < log(1.26).
By repeating this process we can muddle our way to a decent approximation.
The value is not rational but here are 20 decimals: log(x)=1.2589254117941672104
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