Logs with bases that are powers of one base: convert to a common base

CAT has tested this idea 5 times across 5 papers, from 2017 to 2025. Solve one, and the others are the same method with different numbers or wording, which is why repeated concepts are the most efficient thing to practise.

Question	What it asks	Answer
CAT 2024 Slot 1 · QA Q22	log_10, log_100, log_1000 of x	31
CAT 2023 Slot 2 · QA Q5	log_√3 x plus a ratio of logs base x	7
CAT 2025 Slot 2 · QA Q2	bases 64, 8, 512; then minimise x + y + z	B
CAT 2017 Slot 1 · QA Q23	log_0.008 √5 + log_√3 81	C
CAT 2024 Slot 2 · QA Q9	log_8 and log_27 over log_2 and log_3	14

Part of Logarithms & Exponents (55 questions since 2017).