(i) Start with the equation \(\log_2(x+5) = 5 - \log_2 x\).

Using the law of logarithms, \(\log_2(x+5) + \log_2 x = 5\).

This can be rewritten as \(\log_2((x+5)x) = 5\).

Therefore, \((x+5)x = 2^5\).

So, \(x^2 + 5x = 32\).

Rearranging gives the quadratic equation \(x^2 + 5x - 32 = 0\).

(ii) Solve the quadratic equation \(x^2 + 5x - 32 = 0\).

Using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 5\), \(c = -32\).

Calculate the discriminant: \(b^2 - 4ac = 5^2 - 4 \times 1 \times (-32) = 25 + 128 = 153\).

Thus, \(x = \frac{-5 \pm \sqrt{153}}{2}\).

The positive solution is \(x = \frac{\sqrt{153} - 5}{2}\), which is approximately \(x = 3.68\).

(i) Start with the equation \(\log_{10}(x+5) = 2 - \log_{10} x\).

Use the property of logarithms: \(\log_{10} a - \log_{10} b = \log_{10} \left( \frac{a}{b} \right)\).

Rearrange the equation: \(\log_{10}(x+5) + \log_{10} x = 2\).

Combine the logarithms: \(\log_{10}((x+5)x) = 2\).

Convert the logarithmic equation to an exponential equation: \((x+5)x = 10^2\).

Simplify: \(x^2 + 5x = 100\).

Thus, the quadratic equation is \(x^2 + 5x - 100 = 0\).

(ii) Solve the quadratic equation \(x^2 + 5x - 100 = 0\) using the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \(a = 1\), \(b = 5\), and \(c = -100\).

Calculate the discriminant: \(b^2 - 4ac = 5^2 - 4 \times 1 \times (-100) = 25 + 400 = 425\).

Find the roots: \(x = \frac{-5 \pm \sqrt{425}}{2}\).

Since \(x\) must be positive, choose the positive root: \(x = \frac{\sqrt{425} - 5}{2}\).

Approximate the value: \(x \approx 7.81\).