Question

Find the mantissa of the logarithm of the number \[0.002359\].
A. \[3710\]
B. \[3718\]
C. \[3728\]
D. \[3742\]

Answer 1

Hint: In this question we need to find the mantissa value of \[0.002359\]. For this purpose we should have a log table with us. Logarithm is the inverse operation to exponentiation. log generally refers to a logarithm base \[10\]. We can see that there are three main columns, first column represent the base \[10\] value from \[10\] to \[99\], second column represents the value of the immediate digit after the decimal point and the third column represents the value of the second digit from the decimal point. The result of log has two parts: one is characteristic or magnitude and the other is called mantissa which is irrespective of the magnitude.

Complete step by step answer:
We know that the log table value starts from \[10\], but our given value is in decimal \[0.002359\] for the convenience of changing it to the standard form and dividing it by \[100000\].
\[0.002359 = \dfrac{{23.59}}{{100000}} \\
\Rightarrow 0.002359= 23.59 \times {10^{ - 5}}\]
Now we have the value \[23.59 \times {10^{ - 5}}\], here the characteristic is \[ - 5\].In the log table note the value of the first column direct to \[23\]. Then in the second column look for the value of \[5\], which is \[3711\].

Now at last in the third column mean difference note the value corresponding to \[9\], we can see the value as \[17\]. Now add the values we found. \[3711 + 17 = 3728\], this is the required mantissa. Thus \[\log (23.59 \times {10^{ - 5}}) = \overline 5 .3728\]. We already discussed that the result of the log has two parts. The characteristic represents the magnitude of base \[10\].

Hence option C is correct.

Note: As mantissa is irrespective of magnitude the mantissa value of \[0.002359\], \[0.02359\], \[2.359\] and \[23.59\] will not be change only their value to the base \[10\] will change which results in their characteristic value that we can see in the log value. Thus \[\log (0.002359) = \overline 3 .3278\], \[\log (0.02359) = \overline 2 .3278\] and \[\log (0.2359) = \overline 1 .3278\] all here the mantissa is the same(value after decimal).