Solve the following problems involving logarithms: a) Find log(6.02214076 x 1023). b) If log x = -1.3649, find i) the characteristic and ii) the mantissa. c) Find the antilog of -1.3649.

Here are the solutions to the problems:

11. Logarithms

a) Use the calculator to find the logarithm of the constant $6.02214076 \times 10^{23}$.
Let $N = 6.02214076 \times 10^{23}$.
We need to find $\log_{10} N$.
Using the logarithm property $\log(AB) = \log A + \log B$:

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

Using a calculator:

$$\log_{10}(6.02214076) \approx 0.779750$$

So,

$$\log_{10} N \approx 0.779750 + 23$$ $$\log_{10} N \approx 23.779750$$

The logarithm of the constant is $\boxed{23.779750}$.

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

i) Characteristics in the logarithm:
To find the characteristic, we express the logarithm in the form $\text{characteristic} + \text{mantissa}$, where the mantissa is positive.

$$-1.3649 = -1 - 0.3649$$

To make the mantissa positive, we add and subtract 1:

$$-1 - 1 + 1 - 0.3649 = -2 + (1 - 0.3649)$$ $$-2 + 0.6351$$

The characteristic is the integer part, which is $-2$. This is often written as $\bar{2}$.
The characteristic is $\boxed{-2}$.

ii) Mantissa in the logarithm:
From the previous step, the mantissa is the positive decimal part.
The mantissa 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 $-1.3649 = \bar{2}.6351$.
To find the antilog of $\bar{2}.6351$:

1. Look up the mantissa 0.6351 in the antilogarithm tables.
   • Antilog of 0.635 is 4.315.
   • The mean difference for 1 (in the row for 63) is 1.
   • So, the antilog of 0.6351 is $4.315 + 0.001 = 4.316$.
2. The characteristic is $\bar{2}$, which means the decimal point should be placed such that there is one zero immediately after the decimal point before the first significant digit. (Alternatively, $10^{\bar{2}.6351} = 10^{-2} \times 10^{0.6351} = 0.01 \times 4.316$). Therefore, the value is 0.04316. The value of $10^{-1.3649}$ is $\boxed{0.04316}$.

12. A grocery store has fruits in racks as shown in the table.
Taking $x$ to represent banana and $y$ to represent orange, work out a simplified expression for the total number of fruits in the store.

From the table:
Number of bananas in Rack 1 = $a$
Number of bananas in Rack 2 = $a$
Total number of bananas = $a + a = 2a$.

Number of oranges in Rack 1 = $b$
Number of oranges in Rack 2 = $b$
Total number of oranges = $b + b = 2b$.

The total number of fruits in the store is the sum of total bananas and total oranges.
Total fruits = $2a + 2b$.
This expression can be simplified by factoring out the common factor of 2.
Total fruits = $2(a+b)$.
The simplified expression for the total number of fruits is $\boxed{2(a+b)}$.

13. A carpenter at Gikomba furniture shop is making a rectangular table top. The length is $(x+5)$ cm and the width is $(x-2)$ cm. Form a simplified expression for the area of the table top.

The area of a rectangle is given by the formula $A = \text{length} \times \text{width}$.
Given:
Length $L = (x+5)$ cm
Width $W = (x-2)$ cm

Substitute these into the area formula:

$$A = (x+5)(x-2)$$

Expand the expression by multiplying the terms:

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