Use logarithmic laws and definitions  log𝑎 (𝑥) + log𝑎(𝑦) = log𝑎 (𝑥𝑦)  log𝑎 (𝑥) − log𝑎(𝑦) = log𝑎 (𝑥 𝑦)  log𝑎 (𝑥𝑛) = 𝑛 log𝑎 (𝑥)  log𝑎 (𝑥) = log𝑏(𝑥) log𝑏(𝑎)  log𝑎 (𝑎) = 1  log𝑎 (1) = 0

Solve for $x$ in the equation $\log_5(x) + \log_5(2) = \log_5(18)$.

The concentration of a chemical compound in a solution decreases over time according to the model $C(t) = 50 \times 10^{-0.2t}$, where $C$ is the concentration in mg/L and $t$ is the time in hours. (a) Use logarithmic laws to find the exact time when the concentration reaches 5 mg/L. Show all algebraic working. (2 marks) (b) Hence, determine the time to the nearest hour. (1 mark)

A data analyst is comparing two growth models for online user engagement. Model P predicts that user count will reach 15,625 after a certain period, while Model Q predicts 78,125 users at the same time. Both models use base-5 logarithmic scaling for their projections. The analyst needs to express the combined prediction as a single logarithmic value. Determine the value of \( \log_5(15{,}625) + \log_5(78{,}125) \), expressing your answer as an integer.