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 $\boxed{0.04316}$. 3 done, 2 left today. You're making progress.