Question

Find the antilogarithm of each of the following:
i. $2.7523$
ii. $3.7523$
iii. $5.7523$
iv. $0.7523$
v. $\overline{1}.7523$
vi. $\overline{2}.7523$
vii. $\overline{3}.7523$

Answer 1

i. $antilog\left(2.7523\right)$

This means logarithm of some number $x$ is $\mathrm{2.7523.}$

Here the characteristic is $2$.

The mantissa of $2.7523$ is positive and is equal to $0.7523$.

Now, look into the row starting $0.75$. In this row, look at number in the column headed by $2$. The number is $5649$.

Now in the same row move in the column of mean differences and look at the number in the column headed by $3$. The number there is $4$.

Add this number to $5649$, we get $5653$.

Since, the characteristic is $2$. So the decimal point is put after three digits, we get $565.3$.

Hence, $antilog\left(2.7523\right)=565.3$

ii. $antilog\left(3.7523\right)$

Here, the characteristic is $3$

The mantissa of $3.7523$ is the same as the mantissa of the number in step (i) i.e $5653$

The decimal point is put after four digits, we get $\mathrm{5653.0.}$

Hence, $antilog\left(3.7523\right)=5653.0$

iii. $antilog\left(5.7523\right)$

Here, the characteristic is $5$.

The mantissa of $5.7523$ is the same as the mantissa of the number in step (i) i.e. $5653$

The decimal point is put after six digits, we get $565300.0$

Hence, $antilog\left(5.7523\right)=565300.0$

iv. $antilog\left(0.7523\right)$

Here, the characteristic is $0$.

The mantissa of $0.7523$ is the same as the mantissa of the number in step (i) i.e. $5653$

The decimal point is put after 1 digit, we get $5.653$

Hence, $antilog\left(0.7523\right)=5.653$.

v. $antilog\left(\overline{1}.7523\right)$

In this case, the characteristic is $\overline{1}$, i.e. -1.

The mantissa of $0.7523$ is the same as the mantissa of the number in step (i) i.e. $5653$

Since, the characteristic is negative 1, the decimal point is to be moved in right $(-1+1=0)$ digit, we get $0.5653$

Hence, $antilog\left(\overline{1}.7523\right)=0.5653$

vi. $antilog\left(\overline{2}.7523\right)$

In this case, the characteristic is $\overline{2}$, i.e. $-2$.

The mantissa of $0.7523$ is the same as the mantissa of the number in step (i) i.e. $5653$

In this, case, the characteristic is $\overline{2}$, i.e.-2, the decimal point is to be moved in one digit $(-2+1=-1)$ in right .

So, we write one zero on the right side of the decimal point.

Hence, $antilog\left(\overline{2}.7523\right)=0.05653$

vii. $antilog\left(\overline{3}.7523\right)$

In this case, the characteristic is $\overline{3}$, i.e. -3.

The mantissa of $0.7523$ is the same as the mantissa of the number in step (i) i.e. $5653$

In this, case, the characteristic is $\overline{3}$, i.e.-3, the decimal point is to be moved in 2 digits $(-3+1=-2)$ in right .

So, we write 2 zeros on the right side of the decimal point.

Hence, $antilog\left(\overline{3}.7523\right)=0.005653$.