CAT 2025 Logarithms Question (Hard)

Question

If , where and are positive real numbers, then the minimum possible value of is

Accepted Answer

64 8 2 512 8 3 log_ 64 x 2 log_ 8 3 log_ 512 z 4, Property of log log_ b m a n m log_ba log_ 8 2 x 2 log_ 8 3 log_ 8 3 z 4 Using the above-mentioned property, the expression becomes log_ 8 x log_ 8 log_ 8 z 4 log_8x cdot z 4 log_8xyz 4 xyz 8 4 2 12 Using AM-GM inequality 3 ge sqrt 3 xyz 3 ge2 4 left

CAT 2025

Quantitative

Logarithms

Hard

If $\log_{64} x^2 + \log_8 \sqrt{y} + 3\log_{512}(\sqrt{y}z) = 4$ , where $x, y$ and $z$ are positive real numbers, then the minimum possible value of $(x+y+z)$ is

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