Question

If the mantissa of $\log 2125 = 3.3275$ , find the mantissa of $\log 21.25$.

Answer 1

Hint:The number given in the decimal form is converted into fractional form and the properties of logarithm are used to evaluate the expression further.

Complete Step-by-step answer:
The mantissa of $\log 2125 = 3.3275$.
The value of $\log 21.25$ is to be found.
$21.25$ can be written as $\dfrac{{2125}}{{100}}$ (After the decimal two digits are there. So, the decimal is removed and a factor of $\dfrac{1}{{100}}$ is multiplied with the number$2125$.
Therefore,
$\log \left( {21.25} \right) = \log \left( {\dfrac{{2125}}{{100}}} \right) \cdots \left( 1 \right)$
Use the property of the logarithm $\log \left( {\dfrac{a}{b}} \right) = \log \left( a \right) - \log \left( b \right)$ in equation (1), the equation becomes as
$\Rightarrow\log 21.25 = \log 2125 - \log 100 \cdots \left( 2 \right)$
$\Rightarrow\log \left( {100} \right)$ can be written as $\log \left( {{{10}^2}} \right)$ , the equation (2) becomes as
$\Rightarrow\log 21.25 = \log 2125 - \log {10^2} \cdots \left( 3 \right)$
Use the property of logarithm $\log {a^b} = b\log a$ in equation (3),
$\Rightarrow\log 21.25 = \log 2125 - 2\log 10 \cdots \left( 4 \right)$
The value of $\log 10 = 1$ as ${\log _a}a = 1$ and $\log 2125 = 3.3275$ (given in the question), substitute them in equation (4)
\[
\Rightarrow \log 21.25 = 3.3275 - 2\left( 1 \right) \\
\Rightarrow \log 21.25 = 3.3275 - 2 \\
  \Rightarrow\log 21.25 = 1.3275 \\
 \]
The mantissa of $\log 21.25$ is $1.3275$ .

Note:The important points to be remembered are,
$\log 3.134$ can be written as $\log \dfrac{{3134}}{{1000}}$ . This can be written because there are 3 digits after the decimal. So, a factor of $\dfrac{1}{{1000}}$ is multiplied to $3134$.This is done to make the calculation easy.
The value of $\log 12 = 1.079$ . In this expression $1$ is called the characteristic and $0.079$ is called the mantissa.
The following properties of logarithm are important.
$\Rightarrow\log \left( {\dfrac{a}{b}} \right) = \log \left( a \right) - \log \left( b \right)$
For instance, the property is used as
$\Rightarrow\log \dfrac{{3134}}{{1000}} = \log 3134 - \log \left( {1000} \right)$
$\Rightarrow\log {b^c} = c\log b$
For instance, the property is used as,
$
 \Rightarrow \log \left( {1000} \right) = \log \left( {{{10}^3}} \right) \\
  \Rightarrow\log \left( {1000} \right) = 3\log 10 \\
 $
The base of the logarithm can be changed in terms of any other base as,
$\Rightarrow{\log _a}b = \dfrac{{{{\log }_c}b}}{{{{\log }_c}a}}$
For instance, ${\log _2}3$ can be written as
$\Rightarrow{\log _2}3 = \dfrac{{{{\log }_{10}}\left( 3 \right)}}{{{{\log }_{10}}\left( 2 \right)}}$
The base $2$ is converted into base $10$.