Calculate the logarithm of Avogadro's constant.

Step 1: Calculate the logarithm of Avogadro's constant.
Avogadro's constant is given as $6.02214076 \times 10^{23}$.
To find the logarithm, we use the property $\log(A \times B) = \log A + \log B$.

$$\log_{10}(6.02214076 \times 10^{23}) = \log_{10}(6.02214076) + \log_{10}(10^{23})$$ $$\log_{10}(6.02214076 \times 10^{23}) = \log_{10}(6.02214076) + 23$$

Using a calculator for $\log_{10}(6.02214076)$:

$$\log_{10}(6.02214076) \approx 0.779750902$$

Now, add this to 23:

$$23 + 0.779750902 = 23.779750902$$

The logarithm of the constant is $\boxed{23.779750902}$.

b) Abdul works out the logarithm of a number as $-1.3649$.

i) Determine the characteristics in the logarithm.
The characteristic is the integer part of the logarithm. For a negative logarithm, we express it such that the mantissa (fractional part) is positive.

$$-1.3649 = -2 + 0.6351$$

The characteristic is the integer part, which is $-2$. This can also be written as $\bar{2}$.
The characteristics in the logarithm is $\boxed{-2 \text{ or } \bar{2}}$.

ii) Determine the mantissa in the logarithm.
The mantissa is the positive fractional part of the logarithm.

$$-1.3649 = -2 + 0.6351$$

The mantissa is 0.6351.
The mantissa in the logarithm is $\boxed{0.6351}$.

c) Hence use tables of antilogarithms to determine the value of $10^{-1.3649}$.
We need to find the antilogarithm of $-1.3649$.
From part (b), we know that $-1.3649 = \bar{2}.6351$.
This means $10^{-1.3649} = 10^{\bar{2}.6351} = 10^{0.6351} \times 10^{-2}$.
Now, we find the antilog of 0.6351 using antilog tables:

• Look for 0.63 in the rows.
• Look for 5 in the columns, which gives 4315.
• Look for the mean difference for 1, which is 1.
• Add the mean difference: $4315 + 1 = 4316$. So, the antilog of 0.6351 is 4.316. Now, combine this with the characteristic:

$$10^{-1.3649} = 4.316 \times 10^{-2}$$ $$10^{-1.3649} = 0.04316$$

The value of $10^{-1.3649}$ is $\boxed{0.04316}$.