Solve for x; 11. Use logarithms table to evaluate rac86.4 imes 0.410.013

$$x = 2+5$$ $$x = 7$$

The value of $x$ is $\boxed{7}$.

b) $2^{x-1} = 8^{2x+3}$

$$2^{x-1} = 2^{3(2x+3)}$$ $$2^{x-1} = 2^{6x+9}$$

$$x - 6x = 9 + 1$$ $$-5x = 10$$ $$x = \frac{10}{-5}$$ $$x = -2$$

The value of $x$ is $\boxed{-2}$.

11. Use logarithms table to evaluate $\frac{86.4 \times 0.41}{0.013}$
Let $N = \frac{86.4 \times 0.41}{0.013}$.

$$\log 86.4 = 1.9365$$ $$\log 0.41 = \overline{1}.6128 \quad (\text{which is } -1 + 0.6128)$$ $$\log 0.013 = \overline{2}.1139 \quad (\text{which is } -2 + 0.1139)$$

$$\log N = 1.9365 + (\overline{1}.6128) - (\overline{2}.1139)$$ $$\log N = 1.9365 - 1 + 0.6128 - (-2 + 0.1139)$$ $$\log N = 1.9365 - 1 + 0.6128 + 2 - 0.1139$$ $$\log N = (1 - 1 + 2) + (0.9365 + 0.6128 - 0.1139)$$ $$\log N = 2 + 1.4354$$ $$\log N = 3.4354$$

$$N = \text{antilog}(3.4354)$$ $$N = 10^{3.4354}$$ $$N \approx 2725$$

The evaluated value is $\boxed{2725}$.

12. Evaluate the following logarithms

a) $\overline{3}.892 \times 2$

$$= (-3 \times 2) + (0.892 \times 2)$$ $$= -6 + 1.784$$

$$= -6 + 1 + 0.784$$ $$= -5 + 0.784$$ $$= \overline{5}.784$$

The evaluated logarithm is $\boxed{\overline{5}.784}$.

b) $\overline{2}.106 \div 2$

$$= \frac{-2}{2} + \frac{0.106}{2}$$ $$= -1 + 0.053$$

$$= \overline{1}.053$$

The evaluated logarithm is $\boxed{\overline{1}.053}$.