Basic Mathematics & Logarithm Question - None

Question

Let $x = (\text{antilog}_2 3) \cdot \log_3 2$, $y = \log_2(\log_3(\log_2 512))$ and $z = \log_5 3 \cdot \log_7 5 \cdot \log_2 7$, then $xyz$ is equal to:

Accepted Answer

Step 1: Calculate x

x = (antilog₂ 3) · log₃ 2 = 2³ · log₃ 2 = 8 · log₃ 2

Using the identity log_a(b) · log_b(a) = 1:

log₃ 2 · log₂ 3 = 1, so log₃ 2 = 1/log₂ 3

Therefore: x = 8 · (1/log₂ 3) = 8/log₂ 3

Step 2: Calculate y

First: log₂ 512 = log₂ 2⁹ = 9

Then: log₃(log₂ 512) = log₃ 9 = log₃ 3² = 2

Therefore: y = log₂(log₃(log₂ 512)) = log₂ 2 = 1

Step 3: Calculate z

z = log₅ 3 · log₇ 5 · log₂ 7

Using the chain rule: log_a(b) · log_b(c) · log_c(d) = log_a(d)

Reorder: z = log₅ 3 · log₅ 7 · log₇ 2... Actually, apply directly:

z = log₂ 3 (using the property that log₅ 3 · log₇ 5 · log₂ 7 = log₂ 3 via change of base)

Step 4: Calculate xyz

xyz = (8/log₂ 3) · 1 · log₂ 3 = 8

∴ Answer: 8

Question:

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