Let me help you break down and solve the problems in the image.

1. (a) Solve the logarithmic equations:

• $\log(x+1) - \log x = \log(4x)$

Solution:

Using the property of logarithms: $\log(a) - \log(b) = \log\left(\frac{a}{b}\right)$ We get: $\log\left(\frac{x+1}{x}\right) = \log(4x)$ Since the logarithms on both sides are equal, we can equate the arguments: $\frac{x+1}{x} = 4x$ Multiply both sides by $x$: $x+1 = 4x^2$ Rearrange: $4x^2 - x - 1 = 0$ This is a quadratic equation. Solve using the quadratic formula: $x = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(4)(-1)}}{2(4)} = \frac{1 \pm \sqrt{1+16}}{8} = \frac{1 \pm \sqrt{17}}{8}$ The solutions are: $x = \frac{1 + \sqrt{17}}{8} \quad \text{or} \quad x = \frac{1 - \sqrt{17}}{8}$ Since logarithms require positive values, discard the negative solution.

Thus, $x = \frac{1 + \sqrt{17}}{8}$.

1. (b) Solve the logarithmic equation:

$3\log x + \log 64 = 11$

Solution:

First, combine the logarithms using the logarithmic property: $3\log x = \log x^3$ Now the equation becomes: $\log x^3 + \log 64 = 11$ Using the property $\log a + \log b = \log(ab)$: $\log(64x^3) = 11$ Convert the logarithmic equation to exponential form: $64x^3 = 10^{11}$ Divide both sides by 64: $x^3 = \frac{10^{11}}{64}$ Take the cube root of both sides: $x = \sqrt[3]{\frac{10^{11}}{64}}$

2. (a) Solve the system of simultaneous equations: