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The mantissa of log7623 is .8821259; write down the logarithms of 7.623, 762.3, .007623, 762300, .000007623.

Answer
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Hint: Here, we need to solve the logarithm of decimal numbers. We can use the formula:
logx= characteristics + mantissa

Complete step-by-step answer:
We will find the characteristics first and then use the mantissa value given in the question.
Take 7.623,
1<7.623<10
Taking logarithm on both sides, (using the rule of sandwich theorem)
log1<log7.623<log10
0<log7.623<1
We found that the characteristic of log 7.623 is 0
So from sandwich theorem it is clear that the value of log 7.623 lies in between 0 and 1.
From the given information we know that mantissa of log 623 is 0.8821259 (or we can use log table)
Then the value of log 7.623 = characteristic + mantissa
log7.623 = 0 + .8821259 = 0.8821259.
Here we use the value of log 7.623 to find remaining values or we can repeat the above procedure.
log762.3 = log(7.623×102)= 2 + 0.8821259 = 2.8821259
log0.007623 = log(7.623×103)= 0.8821259 – 3 = -2.1178741
log762300 = log(7.623×105)= 5 + 0.8821259 = 5.8821259
log0.000007623= log (7.623×106)=0.8821259 - 6 = -5.1178741

Note: Integer portion of the value of a logarithm is called characteristic and the fractional part (part of number which is located after decimal point) is called mantissa. For example log 27=1.431, here 1 is the characteristic and 0.431 is mantissa. To find the logarithmic value of a decimal number (ex.762.3) we have to find the characteristic and mantissa. Then adding both of them will give the value of a decimal number.
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