QUESTION

# The mantissa of $\log 7623$ is .8821259; write down the logarithms of 7.623, 762.3, .007623, 762300, .000007623.

Hint: Here, we need to solve the logarithm of decimal numbers. We can use the formula:
$\log x =$ characteristics + mantissa

We will find the characteristics first and then use the mantissa value given in the question.
Take 7.623,
$\Rightarrow 1 < 7.623 < 10$
Taking logarithm on both sides, (using the rule of sandwich theorem)
$\Rightarrow \log 1 < \log 7.623 < \log 10$
$\Rightarrow 0 < \log 7.623 < 1$
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
$\Rightarrow \log$7.623 = 0 + .8821259 = 0.8821259.
Here we use the value of log 7.623 to find remaining values or we can repeat the above procedure.
$\log 762.3$ = $\log \left( {7.623 \times {{10}^2}} \right) =$ 2 + 0.8821259 = 2.8821259
$\log 0.007623$ = $\log \left( {7.623 \times {{10}^{ - 3}}} \right) =$ 0.8821259 – 3 = -2.1178741
$\log 762300$ = $\log \left( {7.623 \times {{10}^5}} \right) =$ 5 + 0.8821259 = 5.8821259
$\log 0.000007623$= $log{\text{ }}\left( {7.623 \times {{10}^{ - 6}}} \right) =$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.